Decision making, chaos and determinism

Dr. Gerald H Thomas

MSOE

Introduction

Decision making is one of the most human acts and seems to be the most difficult area to formalize into a theory of behaviors that are causal and deterministic. In fact one might think that the very nature of decision making is one of chance and uncertainty. One issue we think relevant is the general lack of understanding of causal theories and how they deal with uncertainty. Moreover, in our view, there is insufficient appreciation of the sensitivity of the initial conditions that determine future behaviors. When these issues are taken into account, it becomes easier to see the possibility for causal formal theories of human decision making. See related and much earlier MIT paper on this same subject based on Systems Dynamics, http://dspace.mit.edu/bitstream/handle/1721.1/2336/SWP-3241-23273135.pdf?sequence=1, Sterman et al. 

One thing that might help our understanding of this issue is to explore how we deal with uncertainty in such mundane activities as measurements, which form the basis of physical theories. So for example we understand by length, an attribute that characterizes the height, width or depth of something. It is a great accomplishment in understanding to separate this concept from the mechanisms by which we perform the measurement. The mechanisms involve a measuring stick and us as agent; for those of you measuring a basement, you know that multiple measurements yield multiple answers. Yet we are confident that the basement has well defined dimensions. How did we come to this conclusion and how did we learn to separate out the uncertainties associated with us as agent and the intermediary of a measurement stick, from the invariant concept of length? I think we all agree that the separation has been done and we are comfortable with the idea of length.

Decision making

What part is certain and what part is uncertain?

Determinism

What are the characteristics we expect of activities that are determined? Determined activities differ from stochastic activities.

Chaos

Deterministic but the results depend sensitively on the starting point

Elasticity

The system is more than deterministic and occasionally chaotic: it is also elastic. Neighbor events impact each other and are not independent! We will investigate decision process theory that has all of these characteristics and roughly can be thought of as producing behavior that follows the following equations which are similar to the motion of charged particles in an electromagnetic field

Stock Market--chaotic decision making on a large scale?

Get data using Wolfram Alpha

mem=FinancialData["^DJI","Members"]

listdata=First[FinancialData[#,"Price",{{2000},{2013},"Month"}][[All,2]]&/@mem[[-1;;-1]]]

Recurrence plot of stock market data

listArray=Table[Abs[listdata[[i]]-listdata[[j]]],{i,150},{j,150}];

ListContourPlot[listArray,ColorFunction->Hue,Contours->20]

Determinism: physics

Physics example: the pendulum

We believe we understand how a pendulum works because we can predict the behavior starting from a description in which we describe its restoring force  as being the source of the acceleration.

Manipulate[Initialization[Module[{q0,p0,b,t},eqn={p'[t]==-Sin[q[t]]+b Sin[f t],q'[t]==p[t],q[0]==qz,p[0]==pz};sol=ParametricNDSolve[eqn,{p,q},{t,0,500},{qz,pz,b,f}];pN[qz_,pz_,b_,f_][t_]:=p[qz,pz,b,f][t]/.sol;qN[q0_,p0_,b_,f_][t_]:=q[q0,p0,b,f][t]/.sol;]];MatrixForm[{{ParametricPlot[{Sin[qN[q0,p0,βp,ν][t]],pN[q0,p0,βp,ν][t]},{t,0,tmax},ImageSize->imageSize,AspectRatio->1,AxesLabel->{"sin q","p"}],ContourPlot[Abs[pN[q0,p0,βp,ν][t]-pN[q0,p0,βp,ν][tPrime]],{t,0,tmax},{tPrime,0,tmax},ImageSize->imageSize,Contours->100
,ColorFunction->Hue,PlotLabel->"p recurrence plot"]},{Plot[qN[q0,p0,βp,ν][t],{t,0,tmax},ImageSize->imageSize,PlotStyle->Blue,AxesLabel->{"t","q"}],Plot[pN[q0,p0,βp,ν][t],{t,0,tmax},ImageSize->imageSize,PlotStyle->Red,AxesLabel->{"t","p"}]},{Plot[Sin[qN[q0,p0,βp,ν][t]],{t,0,tmax},ImageSize->imageSize,PlotStyle->Blue,AxesLabel->{"t","sin q"}],ContourPlot[Abs[Sin[qN[q0,p0,βp,ν][t]]-Sin[qN[q0,p0,βp,ν][tPrime]]],{t,0,tmax},{tPrime,0,tmax},ImageSize->imageSize,Contours->100,ColorFunction->Hue,PlotLabel->"sin q recurrence plot"]}}],{{q0,0,"(Subscript[q, 0])"},-Pi,Pi},{{p0,0,"(Subscript[p, 0])"},-4,4},{{βp,1,"β driver amplitude"},-2,2},{{ν,1,"ν driver frequency"},0,2Pi},{{tmax,15},.1,500},{{imageSize,250},100,500},SaveDefinitions->True]

Determinism: decision process theory

Game theory was one of the first to make the split between what is preserved and what is uncertain. We don’t know which choice a person makes, but we can determine the frequency of choices made from a selection of pure strategies. The zero-sum rock-scissors-paper game children play illustrates the vast complexity of possibilities.

Rock paper Scissors analyzed: linear programming (game theory)

Input

We start with an input game matrix G :

value=1/10
GameScale=1
LabelOne={"rock","scissors","paper"}
LabelTwo={"rock","scissors","paper"}
LabelComposite=Join[LabelTwo,LabelOne,{"time"}]
MatrixForm[G=GameScale (value+{{0,1,-1},{-1,0,1},{1,-1,0}}),TableAlignments->{Right,Right},TableHeadings->{LabelOne,LabelTwo}]
MatrixForm[(G-value)/GameScale,TableAlignments->{Right,Right},TableHeadings->{LabelOne,LabelTwo}]

block matrix

We construct a block matrix that is symmetric

The block will be made up of the input matrix, as well as a zero matrix and unit payoff matrices such that the new game is essentially choose black, white, or hedge with payoffs such that player one gets 1 unit if he/she hedges and player 2 chooses white, player 1 gets one unit if he/she chooses white and player 2 chooses black, and zero if both choose white. The game so constructed is to be symmetric. The elements then are

number of strategies for player 1 and 2

mmInputMatrix = Length[G]

nnInputMatrix = Length[Last[G]]

strategic gap

MaxMin = Max[Table[Min[G[[i]]], {i, mmInputMatrix}]]

MinMax = Min[Table[Max[Transpose[G][[i]]], {i, nnInputMatrix}]]

MaxMin <= actualvalue <= MinMax

strategicGap = MinMax - MaxMin

Self Payoff and Valuation blocks

Player 2 self energy block

nnzero=ConstantArray[0,{nnInputMatrix,nnInputMatrix}]
MatrixForm[nnzero=0{{0,1,0},{-1,0,0},{0,0,0}}]

Player 1 self energy block

mmzero=ConstantArray[0,{mmInputMatrix,mmInputMatrix}]
MatrixForm[mmzero=0{{0,0,0},{0,0,1},{0,-1,0}}]

Player 2 valuation field

nnone = {Table[1+0/100, {i, nnInputMatrix}]}

Player 1 valuation field

mmone = {Table[1-0/100, {i,mmInputMatrix}]}

symmetrized matrix F

MatrixForm[symInputMatrix = -ArrayFlatten[{{nnzero, Transpose[G], -Transpose[nnone]}, {-G, mmzero, Transpose[mmone]}, {nnone, -mmone, {{0}}}}]]
addValue=Max[Flatten[symInputMatrix]]

Analysis of symmetrized matrix

The most negative element is

-addValue

The payoff matrix is derived from this such that all elements are positive or zero; the payoff matrix is for player 1:

MatrixForm[rpayoff = symInputMatrix +addValue]

To put the problem in linear programming form for Mathematica, we minimize c.x and introduce the vector b that sets payoff.x >= b, which is the constraint

b = Table[1, {i,Length[rpayoff]}]

and the vector c that defines the minimization quantity c.x

c = Table[1, {i,Length[Last[rpayoff]]}]

Solution for player 2 and player 1

strategies

Using these quantities, the solution is provided by  using the linear programming built in function:

x = LinearProgramming[c, rpayoff, b]

The value, in the symmetrized game,  of payoff for player 1 is. For the original game we must have v-addValue, which we expect to be 0:

v = 1/c.x
v-addValue

and the strategy vector is (the first nnInputMatrix elements  are player 2, and the next mmInputMatrix elements are player 1)

strategy = v x
symInputMatrix.strategy

The individual strategies for player 1 and player 2 can then be extracted, normalized to unity:

strategyplayer1 = Table[strategy[[i+nnInputMatrix]], {i, mmInputMatrix}]/Sum[strategy[[j+nnInputMatrix]], {j,mmInputMatrix}]

strategyplayer2 = Table[strategy[[i]], {i, nnInputMatrix}]/Sum[strategy[[j]], {j, nnInputMatrix}]

Direct Approach

Just compute the null space of the input matrix: with valuation fields, you may get negative strategies

y=First[NullSpace[symInputMatrix]]
If[y[[7]]==0,y,y=y/y[[7]]]
If [x[[Length[symInputMatrix]]]==0,x,x/x[[Length[symInputMatrix]]]]

Sensitivity analysis

MatrixForm[symInputMatrix]
MatrixForm[G-value]
MatrixForm[negotiation[rn1_,rn2_,rn3_,rn4_,rn5_,rn6_,rn7_,rn8_,nr9_,rn10_]={{rn7,rn1,-rn2},{-rn3,rn8,rn4},{rn5,-rn6,rn9}}+rn10]
MatrixForm[selfNeg1[rn1_,rn2_,rn3_]=-{{0,rn1,rn2},{-rn1,0,rn3},{-rn2,-rn3,0}}]
MatrixForm[selfNeg2[rn1_,rn2_,rn3_]=-{{0,rn1,rn2},{-rn1,0,rn3},{-rn2,-rn3,0}}]
nn[rn1_,rn2_,rn3_,rn4_]={{rn1+value/rn4,rn2+value/rn4,rn3+value/rn4}}
mm[rn1_,rn2_,rn3_,rn4_]={{rn1+value/rn4,rn2+value/rn4,rn3+value/rn4}}
MatrixForm[F[rn1_,rn2_,rn3_,rn4_,rn5_,rn6_,rn7_,rn8_,rn9_,rn10_,rn11_,rm1_,rm2_,rm3_,rm4_,rm5_,rm6_,rp1_,rp2_,rp3_,rp4_,rp5_,rp6_]=-ArrayFlatten[{{selfNeg2[rp4,rp5,rp6],Transpose[negotiation[rn1,rn2,rn3,rn4,rn5,rn6,rn7,rn8,rn9,rn10]],-Transpose[nn[rm4,rm5,rm6,rn11]]},{-negotiation[rn1,rn2,rn3,rn4,rn5,rn6,rn7,rn8,rn9,rn10],selfNeg1[rp1,rp2,rp3],Transpose[mm[rm1,rm2,rm3,rn11]]},{nn[rm4,rm5,rm6,rn11],-mm[rm1,rm2,rm3,rn11],{{0}}}}]]

Decision process theory model (an extension of game theory)

MatrixForm[symInputMatrix]
Manipulate[MatrixForm[{tmp1=Chop[NullSpace[tmp=F[rn1,rn2,rn3,rn4,rn5,rn6,rn7,rn8,rn9,rn10,rn11,rm1,rm2,rm3,rm4,rm5,rm6,rp1,rp2,rp3,rp4,rp5,rp6]]],tmp2=First[tmp1],If[tmp2[[7]]==0,tmp3=tmp2,tmp3=tmp2/tmp2[[7]]],{ps1=Sum[tmp3[[i]],{i,4,6}],Table[tmp3[[i]]/ps1,{i,4,6}]},{ps2=Sum[tmp3[[i]],{i,3}],Table[tmp3[[i]]/ps2,{i,3}]},MatrixForm[tmp]},TableSpacing-> 10,TableAlignments->Left,TableHeadings->{{"Null Vectors","Player 1 frequencies", "Normalized frequencies","player 1 sum and strategies","player 2 sum and strategies","player 1 Payoff Matrix"},None}],{{rn1,1,"p1:rock beats scissors"},-2,4},{{rn2,1,"p1:rock loses to paper"},-2,4},{{rn3,1,"p1:scissors broken by rock"},-2,4},{{rn4,1,"p1:scissors cuts paper"},-2,4},{{rn5,1,"p1:paper covers rock"},-2,4},{{rn6,1,"p1:paper cut by scissors"},-2,4},{{rn7,0,"p1:rock tie"},-2,2},{{rn8,0,"p1:scissors tie"},-2,2},{{rn9,0,"p1:paper tie"},-2,2},{{rn10,value,"p1:CollectiveBiasValue"},-4+value,2+value},{{rn11,1,"p1:inertia"},1/10,100},{{rp1,0,"p1 conflict: rock->scissors"},-2,2},{{rp2,0,"p1 conflict: rock->paper"},-2,2},{{rp3,0,"p1 conflict: scissors->paper"},-2,2},{{rp4,0,"p2 conflict: rock->scissors"},-2,2},{{rp5,0,"p2 conflict: rock->paper"},-2,2},{{rp6,0,"p2 conflict: scissors->paper"},-2,2},{{rm1,0,"p1 values rock"},-2,2},{{rm2,0,"p1 values scissors"},-2,2},{{rm3,0,"p1 values paper"},-2,2},{{rm4,0,"p2 values rock"},-2,2},{{rm5,0,"p2 values scissors"},-2,2},{{rm6,0,"p2 values paper"},-2,2},SaveDefinitions->True]

Elasticity is a dynamic extension

Models in decision process theory take some time to compute; the basic types of results can be illustrated by a simple toy model that depends on one strategy (in the elastic direction) z and time t. The following is a recurrence plot in time for different strategies z. Stock market shows some similarities to these plots.

Toy network model--recurrence plot

g[z_,t_]:=Cos[z]Cos[t]+Sin[2t]Sin[3z]
Manipulate[ContourPlot[Evaluate[Abs[g[z,t]-g[z,tPrime]]],{t,0,10},{tPrime,0,10},ColorFunction->Hue,Contours->100,ImageSize->500],{{z,0},-5,5},SaveDefinitions->True]

Chaos, determinism and elasticity

Decision process theory will have non-linear effects just like the pendulum problem. These produce chaotic behaviors, where the results depend sensitively on the starting point.

We said: The system is more than deterministic and occasionally chaotic: it is also elastic. Neighbor events impact each other and are not independent! We will investigate decision process theory that has all of these characteristics and roughly can be thought of as producing behavior that follows the following equations which are similar to the motion of charged particles in an electromagnetic field



Frame dependent rotational effects: gradient forces (centripetal and gravitational forces) and velocity dependent forces (Coriolis forces) come from (Subscript[ω, abc])=(-(∂_a)(Subscript[g, bc])+(∂_b)(Subscript[g, ca])+(∂_c)(Subscript[g, ab]))/2

Attack defense model--two persons each with two strategies

The general form of a dynamic game with two players is the following for player 1, where each player has two strategies, one internal self strategy and two valuations:

MatrixForm[payoff[n1_,n2_,n3_,n4_,s1_,s2_]={{0,s2,-n1,-n3},{-s2,0,-n2,-n4},{n1,n2,0,s1},{n3,n4,-s1,0}}]
payoffValue[v1_,v2_,v3_,v4_]={v3,v4,-v1,-v2}
valuePotential[y1_,y2_,y3_,y4_,v1_,v2_,v3_,v4_]=payoffValue[v1,v2,v3,v4].{y1,y2,y3,y4}
tmin=-300
tmax=300
e1z=(1/2);e2z=(1/2)
initialSpaceFlow[r_,u_,σ1_,σ2_]={((r-u+(Sqrt[2])σ2)/2),((r-u-(Sqrt[2])σ2)/2),((r+u+(Sqrt[2])σ1)/2),((r+u-(Sqrt[2])σ1)/2)}
initialSpaceFlow[1,0,-(1/(Sqrt[2])),-(1/(Sqrt[2]))]
maxInitialTimeFlow=20
initialTimeFlow=4
size[z_,L_]=(L/Pi)Sin[((Pi z)/L)]
ϕ[w_,tL_,α_,β_,γ_,δ_]=(1/2)γ (δ^2)  Log[(w^2)+(δ^2)]+β w Sin[α tL]
MatrixForm[gab=DiagonalMatrix[{-1,-1,-1,-1}]]
g00[w_,tL_,α_,β_,γ_,δ_]=Exp[2ϕ[w,tL,α,β,γ,δ]]
MatrixForm[inverseg=Inverse[gab]]
gradϕ[w_]=Simplify[D[ϕ[w,tL,α,β,γ,δ],w]]
gradtL[w_,tL_,α_,β_,γ_,δ_]=Simplify[D[ϕ[w,tL,α,β,γ,δ],tL]]
gradVector[w_,tL_,α_,β_,γ_,δ_]=gradϕ[w]{-1/2,-1/2,1/2,1/2}
gradForce[w_,tL_,α_,β_,γ_,δ_]=inverseg.gradVector[w,tL,α,β,γ,δ]g00[w,tL,α,β,γ,δ]

Add the characteristic payoff to the mix, and don’t forget to come back and add the value terms as a potential. But first the characteristic payoff:

characteristicPotential[y1_,y2_,y3_,y4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_]=((-1)/2)size[payoff[cn1,cn2,cn3,cn4,cs1,cs2].{y1,y2,y3,y4},αα]
tmp12=D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[2]],y1]-D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[1]],y2]
tmp13=D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[3]],y1]-D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[1]],y3]
tmp14=D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[4]],y1]-D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[1]],y4]
tmp23=D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[3]],y2]-D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[2]],y3]
tmp24=D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[4]],y2]-D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[2]],y4]
tmp34=D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[4]],y3]-D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[3]],y4]
MatrixForm[characteristicPayoff[y1_,y2_,y3_,y4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_]={{0,tmp12,tmp13,tmp14},{-tmp12,0,tmp23,tmp24},{-tmp13,-tmp23,0,tmp34},{-tmp14,-tmp24,-tmp34,0}}]

MatrixForm[payoff[n1,n2,n3,n4,s1,s2]]

We have also set some of the numerical parameters to be used later. We will solve the dynamic equations using parametric solution of the differential equations. As a starting values we choose the values for the prisoner’s dilemma.

inertiaPD=initialTimeFlow
MatrixForm[payoffPD1={{0,0,-(9/10),-1},{0,0,0,-(1/10)},{(9/10),0,0,0},{1,(1/10),0,0}}]
MatrixForm[payoffPD2={{0,0,(9/10),0},{0,0,1,(1/10)},{-(9/10),-1,0,0},{0,-(1/10),0,0}}]
MatrixForm[payoffTotal[n1_,n2_,n3_,n4_,s1_,s2_,y1_,y2_,y3_,y4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,e0_,e1_,e2_]=e1 payoffPD1+e2 payoffPD2+payoff[n1,n2,n3,n4,s1,s2]+e0  characteristicPayoff[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2,αα]]
n1Z=0;n2Z=0;n3Z=0;n4Z=0;s1Z=0;s2Z=0;v1Z=0;v2Z=0;v3Z=0;v4Z=0;rz=1;uz=0;σ1z=-(1/(Sqrt[2]));σ2z=-(1/(Sqrt[2]));αz=1;βz=0;γz=1;δz=1;tLowz=tmin/20;tHighz=tmax/20;cn1Z=0;cn2Z=0;cn3Z=0;cn4Z=0;cs1Z=0;cs2Z=0;ααZ=1;ββZ=1;

valuePotentialPD1=valuePotential[y1,y2,y3,y4,(1/(10inertiaPD)),(1/(10inertiaPD)),(1/(10inertiaPD)),(1/(10inertiaPD))]
valuePotentialPD2=valuePotential[y1,y2,y3,y4,((-1)/(10inertiaPD)),((-1)/(10inertiaPD)),((-1)/(10inertiaPD)),((-1)/(10inertiaPD))]
(*valuePotentialdelta=valuePotential[size[y1,ββ],size[y2,ββ],size[y3,ββ],size[y4,ββ],v1,v2,v3,v4]*)
valuePotentialdelta=size[valuePotential[y1,y2,y3,y4,v1,v2,v3,v4],ββ]
valuePotentialTotal[y1_,y2_,y3_,y4_,v1_,v2_,v3_,v4_]=e1 valuePotentialPD1+e2 valuePotentialPD2+valuePotentialdelta
ValueFieldTotal[y1_,y2_,y3_,y4_,v1_,v2_,v3_,v4_,ββ_]={D[valuePotentialTotal[y1,y2,y3,y4,v1,v2,v3,v4],y1],D[valuePotentialTotal[y1,y2,y3,y4,v1,v2,v3,v4],y2],D[valuePotentialTotal[y1,y2,y3,y4,v1,v2,v3,v4],y3],D[valuePotentialTotal[y1,y2,y3,y4,v1,v2,v3,v4],y4]}

The model will be a dynamic game model with two sets of variables: the flows and the preferences. The payoff matrix and inertia term of the initial flow vector are part of the parameter set. The equation we will solve is the coupled set of equations for the flow vector (x^a)=(V^a): (Subscript[g, ab])((dV^b)/(dτ))=(Subscript[eF, ab])(V^b). Here (Subscript[g, ab]) is a diagonal matrix whose fifth component has the opposite sign of the other components (which are -1) and (Subscript[F, ab]) is the payoff matrix for player 1. All quantities are from that player’s perspective. The preferences are (y^a) and are determined from the flow: ((dy^a)/(dτ))=(V^a). The “time” parameter is the proper time and represents the distance along the path, τ. We start the preference at 0.

var={(Subscript[x, 1]),(Subscript[x, 2]),(Subscript[x, 3]),(Subscript[x, 4]),(Subscript[x, 5]),(Subscript[y, 1]),(Subscript[y, 2]),(Subscript[y, 3]),(Subscript[y, 4]),(Subscript[y, 5])}
flow[t_]={(Subscript[x, 1])[t],(Subscript[x, 2])[t],(Subscript[x, 3])[t],(Subscript[x, 4])[t]}
vBarupperZ[t_]=(((Subscript[x, 5])[t]-valuePotentialTotal[(Subscript[y, 1])[t],(Subscript[y, 2])[t],(Subscript[y, 3])[t],(Subscript[y, 4])[t],v1,v2,v3,v4])/(g00[(((Subscript[y, 3])[t]+(Subscript[y, 4])[t]-(Subscript[y, 1])[t]-(Subscript[y, 2])[t])/2),(Subscript[y, 5])[t],α,β,γ,δ]));
vUpper[t_]=vBarupperZ[t]-characteristicPotential[(Subscript[y, 1])[t],(Subscript[y, 2])[t],(Subscript[y, 3])[t],(Subscript[y, 4])[t],cn1,cn2,cn3,cn4,cs1,cs2].flow[t];
preference[t_]={(Subscript[y, 1])[t],(Subscript[y, 2])[t],(Subscript[y, 3])[t],(Subscript[y, 4])[t]}
parameterList={{n1,-2,2},{n2,-2,2},{n3,-2,2},{n4,-2,2},{s1,-2,2},{s2,-2,2},{v1,-2,2},{v2,-2,2},{v3,-2,2},{v4,-2,2},{cn1,-2,2},{cn2,-2,2},{cn3,-2,2},{cn4,-2,2},{cs1,-2,2},{cs2,-2,2},{αα,1/10,10},{ββ,1/10,10},{inertia,2,maxInitialTimeFlow},{e1,-2,2},{e2,-2,2},{r,-2,2},{u,-2,2},{σ1,-2,2},{σ2,-2,2},{α,0,2Pi},{β,-1,1},{γ,0,2},{δ,0.1,5},{tLow,tmin,0},{tHigh,0,tmax}}
force[t_]=Table[(inverseg.payoffTotal[n1,n2,n3,n4,s1,s2,(Subscript[y, 1])[t],(Subscript[y, 2])[t],(Subscript[y, 3])[t],(Subscript[y, 4])[t],cn1,cn2,cn3,cn4,cs1,cs2,αα,(Subscript[x, 5])[t],e1,e2].flow[t])[[k]],{k,4}];
gradTimeForce1[α_,β_,γ_,δ_,v1_,v2_,v3_,v4_]=gradForce[(((Subscript[y, 3])[t]+(Subscript[y, 4])[t]-(Subscript[y, 1])[t]-(Subscript[y, 2])[t])/2),(Subscript[y, 5])[t],α,β,γ,δ]((vBarupperZ[t])^2)-inverseg.characteristicPotential[(Subscript[y, 1])[t],(Subscript[y, 2])[t],(Subscript[y, 3])[t],(Subscript[y, 4])[t],cn1,cn2,cn3,cn4,cs1,cs2]gradtL[(((Subscript[y, 3])[t]+(Subscript[y, 4])[t]-(Subscript[y, 1])[t]-(Subscript[y, 2])[t])/2),(Subscript[y, 5])[t],α,β,γ,δ]g00[(((Subscript[y, 3])[t]+(Subscript[y, 4])[t]-(Subscript[y, 1])[t]-(Subscript[y, 2])[t])/2),(Subscript[y, 5])[t],α,β,γ,δ]((vUpper[t])^2);
gradTimeForce2[α_,β_,γ_,δ_,v1_,v2_,v3_,v4_]=g00[(((Subscript[y, 3])[t]+(Subscript[y, 4])[t]-(Subscript[y, 1])[t]-(Subscript[y, 2])[t])/2),(Subscript[y, 5])[t],α,β,γ,δ](((vBarupperZ[t])^2)-(((flow[t].characteristicPotential[(Subscript[y, 1])[t],(Subscript[y, 2])[t],(Subscript[y, 3])[t],(Subscript[y, 4])[t],cn1,cn2,cn3,cn4,cs1,cs2]))^2))gradtL[(((Subscript[y, 3])[t]+(Subscript[y, 4])[t]-(Subscript[y, 1])[t]-(Subscript[y, 2])[t])/2),(Subscript[y, 5])[t],α,β,γ,δ];
gradTimeForce3[α_,β_,γ_,δ_,v1_,v2_,v3_,v4_]=inverseg.ValueFieldTotal[(Subscript[y, 1])[t],(Subscript[y, 2])[t],(Subscript[y, 3])[t],(Subscript[y, 4])[t],v1,v2,v3,v4,ββ]vBarupperZ[t];

eqns[t_]=Join[Table[flow'[t][[k]]==force[t][[k]]+gradTimeForce1[α,β,γ,δ,v1,v2,v3,v4][[k]]+gradTimeForce3[α,β,γ,δ,v1,v2,v3,v4][[k]],{k,4}],{(Subscript[x, 5])'[t]==gradTimeForce2[α,β,γ,δ,v1,v2,v3,v4]},Table[preference'[t][[k]]==flow[t][[k]],{k,4}],{(Subscript[y, 5])'[t]==vUpper[t]}];
vBarupperZerotmp=((inertia-valuePotentialTotal[0,0,0,0,v1,v2,v3,v4])/(g00[0,0,α,β,γ,δ]));

initialFlow={initialSpaceFlow[r,u,σ1,σ2][[1]],initialSpaceFlow[r,u,σ1,σ2][[2]],initialSpaceFlow[r,u,σ1,σ2][[3]],initialSpaceFlow[r,u,σ1,σ2][[4]]};
normalizedFlow=initialFlow/(Sqrt[Simplify[initialFlow.gab.initialFlow+g00[0,0,α,β,γ,δ](vBarupperZerotmp^2)]])
vBarupperZero=vBarupperZerotmp/(Sqrt[Simplify[initialFlow.gab.initialFlow+g00[0,0,α,β,γ,δ](vBarupperZerotmp^2)]])
vBarlowerZero=g00[0,0,α,β,γ,δ]vBarupperZero+valuePotentialTotal[0,0,0,0,v1,v2,v3,v4];

initEqns=Simplify[Join[Table[(Subscript[x, k])[0]==normalizedFlow[[k]],{k,4}],Table[(Subscript[y, k])[0]==0,{k,5}],{(Subscript[x, 5])[0]==vBarlowerZero}]];

Differential Equations and parametric solution

We use ParametricNDSolve to obtain the solution:

solution=ParametricNDSolve[Join[eqns[t],initEqns],var,{t,tLow,tHigh},parameterList,MaxSteps->∞]
flowNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=Table[(Subscript[x, k])[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t]/.solution,{k,4}];
preferenceNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=Table[(Subscript[y, k])[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t]/.solution,{k,5}];
flowNum5[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=(Subscript[x, 5])[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t]/.solution;

g00Num[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=g00[(1/2)(preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[3]]+preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[4]]-preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[1]]-preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[2]]),preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[5]],α,β,γ,δ];
valuePotentialNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]= valuePotentialTotal[preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[1]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[2]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[3]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[4]],v1,v2,v3,v4];
vBarupperZNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]= (flowNum5[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t]-valuePotentialNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t])/g00Num[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t];

characteristicPotentialNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=characteristicPotential[preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[1]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[2]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[3]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[4]],cn1,cn2,cn3,cn4,cs1,cs2];
flowNumUpper5[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=vBarupperZNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t]-characteristicPotentialNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t].flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t];

forceNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=inverseg.payoffTotal[n1,n2,n3,n4,s1,s2,preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[1]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[2]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[3]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[4]],cn1,cn2,cn3,cn4,cs1,cs2,αα,flowNum5[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t],e1,e2].flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t];
forceNum5[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=ValueFieldTotal[preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[1]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[2]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[3]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[4]],v1,v2,v3,v4,ββ].flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t];

flowNumComplete[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=Join[flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t],{flowNumUpper5[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t]}];

aNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=valuePotentialNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t];
ϕNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=ϕ[(1/2)(preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[3]]+preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[4]]-preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[1]]-preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[2]]),preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[5]],α,β,γ,δ];
conservedNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=flowNum5[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t];

pathNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=g00Num[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t]((vBarupperZNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t])^2)+flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t].gab.flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t];

transformedFlowNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]={(1/2)(flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[1]]+flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[2]]+flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[3]]+flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[4]]),(1/2)(-flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[1]]-flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[2]]+flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[3]]+flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[4]]),(1/(Sqrt[2]))(flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[3]]-flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t] [[4]]),(1/(Sqrt[2]))(flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[1]]-flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t] [[2]]),flowNumComplete[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[5]]};(*r,u,s1,s2,t*)
spacePayoff=payoffTotal[n1,n2,n3,n4,s1,s2,preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[1]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[2]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[3]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[4]],cn1,cn2,cn3,cn4,cs1,cs2,αα,flowNum5[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t],e1,e2];
timePayoff=ValueFieldTotal[preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[1]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[2]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[3]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[4]],v1,v2,v3,v4,ββ];
combinedPayoff[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]={{spacePayoff[[1]][[1]],spacePayoff[[1]][[2]],spacePayoff[[1]][[3]],spacePayoff[[1]][[4]],timePayoff[[1]]},{spacePayoff[[2]][[1]],spacePayoff[[2]][[2]],spacePayoff[[2]][[3]],spacePayoff[[2]][[4]],timePayoff[[2]]},{spacePayoff[[3]][[1]],spacePayoff[[3]][[2]],spacePayoff[[3]][[3]],spacePayoff[[3]][[4]],timePayoff[[3]]},{spacePayoff[[4]][[1]],spacePayoff[[4]][[2]],spacePayoff[[4]][[3]],spacePayoff[[4]][[4]],timePayoff[[4]]},{-timePayoff[[1]],-timePayoff[[2]],-timePayoff[[3]],-timePayoff[[4]],0}};

inverseg00=1/g00Num[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ]+characteristicPotentialNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ].inverseg.characteristicPotentialNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ];
inversega0=-inverseg.characteristicPotentialNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ];
inversegab[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]={{inverseg[[1]][[1]],inverseg[[1]][[2]],inverseg[[1]][[3]],inverseg[[1]][[4]],inversega0[[1]]},{inverseg[[2]][[1]],inverseg[[2]][[2]],inverseg[[2]][[3]],inverseg[[2]][[4]],inversega0[[2]]},{inverseg[[3]][[1]],inverseg[[3]][[2]],inverseg[[3]][[3]],inverseg[[3]][[4]],inversega0[[3]]},{inverseg[[4]][[1]],inverseg[[4]][[2]],inverseg[[4]][[3]],inverseg[[4]][[4]],inversega0[[4]]},{inversega0[[1]],inversega0[[2]],inversega0[[3]],inversega0[[4]],inverseg00}};
normalg0a=g00Num[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh] [t]characteristicPotentialNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ];
normalgmn=Table[gab[[j]][[k]]+g00Num[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh] [t]characteristicPotentialNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[j]]characteristicPotentialNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[k]],{j,4},{k,4}];
normalgab[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]={{normalgmn[[1]][[1]],normalgmn[[1]][[2]],normalgmn[[1]][[3]],normalgmn[[1]][[4]],normalg0a[[1]]},{normalgmn[[2]][[1]],normalgmn[[2]][[2]],normalgmn[[2]][[3]],normalgmn[[2]][[4]],normalg0a[[2]]},{normalgmn[[3]][[1]],normalgmn[[3]][[2]],normalgmn[[3]][[3]],normalgmn[[3]][[4]],normalg0a[[3]]},{normalgmn[[4]][[1]],normalgmn[[4]][[2]],normalgmn[[4]][[3]],normalgmn[[4]][[4]],normalg0a[[4]]},{normalg0a[[1]],normalg0a[[2]],normalg0a[[3]],normalg0a[[4]],g00Num[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t] }};
deltaab[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=normalgab[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t].inversegab[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t];

Manipulate[{MatrixForm[inversegab[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][tHigh]],MatrixForm[normalgab[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][tHigh]],MatrixForm[Chop[deltaab[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][tHigh]]],
flowNum5[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0],Plot[flowNumUpper5[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ],{t,tLow,tHigh},ImageSize->500]},{{n1,n1Z,"(Subscript[ΔG, 11])"},parameterList[[1]][[2]],parameterList[[1]][[3]]},{{n2,n2Z,"(Subscript[ΔG, 12])"},parameterList[[2]][[2]],parameterList[[2]][[3]]},{{n3,n3Z,"(Subscript[ΔG, 21])"},parameterList[[3]][[2]],parameterList[[3]][[3]]},{{n4,n4Z,"(Subscript[ΔG, 22])"},parameterList[[4]][[2]],parameterList[[4]][[3]]},{{s1,s1Z,"(Subscript[ΔS1, 12])"},parameterList[[5]][[2]],parameterList[[5]][[3]]},{{s2,s2Z,"(Subscript[ΔS2, 12])"},parameterList[[6]][[2]],parameterList[[6]][[3]]},{{v1,v1Z,"(Subscript[Δv, 1])"},parameterList[[7]][[2]],parameterList[[7]][[3]]},{{v2,v2Z,"(Subscript[Δv, 2])"},parameterList[[8]][[2]],parameterList[[8]][[3]]},{{v3,v3Z,"(Subscript[Δv, 3])"},parameterList[[9]][[2]],parameterList[[9]][[3]]},{{v4,v4Z,"(Subscript[Δv, 4])"},parameterList[[10]][[2]],parameterList[[10]][[3]]},{{cn1,cn1Z,"(Subscript[cpG, 11])"},parameterList[[11]][[2]],parameterList[[11]][[3]]},{{cn2,cn2Z,"(Subscript[cpG, 12])"},parameterList[[12]][[2]],parameterList[[12]][[3]]},{{cn3,cn3Z,"(Subscript[cpG, 21])"},parameterList[[13]][[2]],parameterList[[13]][[3]]},{{cn4,cn4Z,"(Subscript[cpG, 22])"},parameterList[[14]][[2]],parameterList[[14]][[3]]},{{cs1,s1Z,"(Subscript[cpS1, 12])"},parameterList[[15]][[2]],parameterList[[15]][[3]]},{{cs2,s2Z,"(Subscript[cpS2, 12])"},parameterList[[16]][[2]],parameterList[[16]][[3]]},{{αα,ααZ,"cpBoxSize"},parameterList[[17]][[2]],parameterList[[17]][[3]]},{{ββ,ββZ,"valueBoxSize"},parameterList[[18]][[2]],parameterList[[18]][[3]]},{{inertia,inertiaPD},parameterList[[19]][[2]],parameterList[[19]][[3]]},{{e1,e1z,"(Subscript[e, 1])"},parameterList[[20]][[2]],parameterList[[20]][[3]]},{{e2,e2z,"(Subscript[e, 2])"},parameterList[[21]][[2]],parameterList[[21]][[3]]},{{r,rz,"(Subscript[V, r])"},parameterList[[22]][[2]],parameterList[[22]][[3]]},{{u,uz,"(Subscript[V, u])"},parameterList[[23]][[2]],parameterList[[23]][[3]]},{{σ1,σ1z,"(Subscript[V, (Subscript[s, 1])])"},parameterList[[24]][[2]],parameterList[[24]][[3]]},{{σ2,σ2z,"(Subscript[V, (Subscript[s, 2])])"},parameterList[[25]][[2]],parameterList[[25]][[3]]},{{α,αz,"driven frequency α"},parameterList[[26]][[2]],parameterList[[26]][[3]]},{{β,βz,"driven amplitude β"},parameterList[[27]][[2]],parameterList[[27]][[3]]},{{γ,γz,"gravity amplitude γ"},parameterList[[28]][[2]],parameterList[[28]][[3]]},{{δ,δz,"gravity range δ"},parameterList[[29]][[2]],parameterList[[29]][[3]]},{{tLow,tLowz,"(Subscript[t, Low])"},parameterList[[30]][[2]],parameterList[[30]][[3]]},{{tHigh,tHighz,"(Subscript[t, High])"},parameterList[[31]][[2]],parameterList[[31]][[3]]}]

Large Format Numerical results with Sensitivity Analysis

The initial payoff matrix is set by the various parameters, as are the eigenvalues and null space vector(s). With the corresponding plot of the time dependence of the strategies, we can see how the eigenvalues are associated with harmonic and exponential behaviors.

Manipulate[MatrixForm[{flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0 ],MatrixForm[combinedPayoff[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0 ]],Chop[Eigenvalues[inversegab[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0 ].combinedPayoff[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0 ]//N]],NullSpace[combinedPayoff[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0 ]],MatrixForm[{{Plot[Evaluate[flowNumComplete[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ]],{t,tLow,tHigh },AxesLabel->{τ,"flow"},PlotStyle->{Green,Purple,Blue,Red,Black},ImageSize->{300},PlotLegends->{"p2:N","p2:C","p1:N","p1:C","timeFlow"}],Plot[Evaluate[transformedFlowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ]],{t,tLow,tHigh },AxesLabel->{τ,"transformed flow"},PlotStyle->{Green,Purple,Blue,Red,Black},ImageSize->{300},PlotLegends->{"r-flow","u-flow","(Subscript[s, 1])-flow","(Subscript[s, 2])-flow","timeFlow"}]},{Plot[Evaluate[{aNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ],characteristicPotentialNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ]}],{t,tLow,tHigh },AxesLabel->{τ,"(Subscript[A, 0])"},PlotStyle->{Black,Blue,Red,Green,Purple},ImageSize->{300},PlotLegends->{"Value Potential","Characteristic Potential 1","Characteristic Potential 2","Characteristic Potential 3","Characteristic Potential 4"}],Plot[Evaluate[conservedNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ]],{t,tLow,tHigh },AxesLabel->{τ,"time-charge"},PlotStyle->Purple,ImageSize->{300},PlotLegends->{"time isometry"}]},{Plot[Evaluate[ϕNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ]],{t,tLow,tHigh },AxesLabel->{τ,"ν"},PlotStyle->Brown,ImageSize->{300},PlotLegends->{"gravity potential"}],Plot[Evaluate[pathNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ]],{t,tLow,tHigh },AxesLabel->{τ,"path length"},PlotStyle->Brown,ImageSize->{300},PlotLegends->{"path"}]}}]},TableSpacing-> 10,TableAlignments->Left,TableHeadings->{{"Flow Vector","Composite Payoff Matrix", "(g^(-1)).F Eigenvalues","Null Space Vector(s)","Plot Flow Vectors"},None}],{{n1,n1Z,"(Subscript[ΔG, 11])"},parameterList[[1]][[2]],parameterList[[1]][[3]]},{{n2,n2Z,"(Subscript[ΔG, 12])"},parameterList[[2]][[2]],parameterList[[2]][[3]]},{{n3,n3Z,"(Subscript[ΔG, 21])"},parameterList[[3]][[2]],parameterList[[3]][[3]]},{{n4,n4Z,"(Subscript[ΔG, 22])"},parameterList[[4]][[2]],parameterList[[4]][[3]]},{{s1,s1Z,"(Subscript[ΔS1, 12])"},parameterList[[5]][[2]],parameterList[[5]][[3]]},{{s2,s2Z,"(Subscript[ΔS2, 12])"},parameterList[[6]][[2]],parameterList[[6]][[3]]},{{v1,v1Z,"(Subscript[Δv, 1])"},parameterList[[7]][[2]],parameterList[[7]][[3]]},{{v2,v2Z,"(Subscript[Δv, 2])"},parameterList[[8]][[2]],parameterList[[8]][[3]]},{{v3,v3Z,"(Subscript[Δv, 3])"},parameterList[[9]][[2]],parameterList[[9]][[3]]},{{v4,v4Z,"(Subscript[Δv, 4])"},parameterList[[10]][[2]],parameterList[[10]][[3]]},{{cn1,cn1Z,"(Subscript[cpG, 11])"},parameterList[[11]][[2]],parameterList[[11]][[3]]},{{cn2,cn2Z,"(Subscript[cpG, 12])"},parameterList[[12]][[2]],parameterList[[12]][[3]]},{{cn3,cn3Z,"(Subscript[cpG, 21])"},parameterList[[13]][[2]],parameterList[[13]][[3]]},{{cn4,cn4Z,"(Subscript[cpG, 22])"},parameterList[[14]][[2]],parameterList[[14]][[3]]},{{cs1,s1Z,"(Subscript[cpS1, 12])"},parameterList[[15]][[2]],parameterList[[15]][[3]]},{{cs2,s2Z,"(Subscript[cpS2, 12])"},parameterList[[16]][[2]],parameterList[[16]][[3]]},{{αα,ααZ,"cpBoxSize"},parameterList[[17]][[2]],parameterList[[17]][[3]]},{{ββ,ββZ,"valueBoxSize"},parameterList[[18]][[2]],parameterList[[18]][[3]]},{{inertia,inertiaPD},parameterList[[19]][[2]],parameterList[[19]][[3]]},{{e1,e1z,"(Subscript[e, 1])"},parameterList[[20]][[2]],parameterList[[20]][[3]]},{{e2,e2z,"(Subscript[e, 2])"},parameterList[[21]][[2]],parameterList[[21]][[3]]},{{r,rz,"(Subscript[V, r])"},parameterList[[22]][[2]],parameterList[[22]][[3]]},{{u,uz,"(Subscript[V, u])"},parameterList[[23]][[2]],parameterList[[23]][[3]]},{{σ1,σ1z,"(Subscript[V, (Subscript[s, 1])])"},parameterList[[24]][[2]],parameterList[[24]][[3]]},{{σ2,σ2z,"(Subscript[V, (Subscript[s, 2])])"},parameterList[[25]][[2]],parameterList[[25]][[3]]},{{α,αz,"driven frequency α"},parameterList[[26]][[2]],parameterList[[26]][[3]]},{{β,βz,"driven amplitude β"},parameterList[[27]][[2]],parameterList[[27]][[3]]},{{γ,γz,"gravity amplitude γ"},parameterList[[28]][[2]],parameterList[[28]][[3]]},{{δ,δz,"gravity range δ"},parameterList[[29]][[2]],parameterList[[29]][[3]]},{{tLow,tLowz,"(Subscript[t, Low])"},parameterList[[30]][[2]],parameterList[[30]][[3]]},{{tHigh,tHighz,"(Subscript[t, High])"},parameterList[[31]][[2]],parameterList[[31]][[3]]}]

it is interesting to see the parametric plot for each player’s flows: to confess versus not to confess.

Manipulate[MatrixForm[{flowNumComplete[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0 ],MatrixForm[combinedPayoff[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0 ]],Chop[Eigenvalues[inversegab[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0 ].combinedPayoff[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0 ]//N]],NullSpace[combinedPayoff[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0 ]],MatrixForm[{{ParametricPlot[{flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[3]],flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[4]]},{t,tLow,tHigh },AxesLabel->{"N","C"},ColorFunction->Function[{t},Hue[t]],PlotStyle->{Thick,Blue},ImageSize->{300},PlotLegends->{"p1"}],ParametricPlot[{flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[1]],flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[2]]},{t,tLow,tHigh },AxesLabel->{"N","C"},ColorFunction->Function[{t},Hue[t]],PlotStyle->{Thick,Green},ImageSize->{300},PlotLegends->{"p2"}]}}]},TableSpacing-> 10,TableAlignments->Left,TableHeadings->{{"Flow Vector","Composite Payoff Matrix", "(g^(-1)).F Eigenvalues","Null Space Vector(s)","Plot Flow Vectors"},None}],{{n1,n1Z,"(Subscript[ΔG, 11])"},parameterList[[1]][[2]],parameterList[[1]][[3]]},{{n2,n2Z,"(Subscript[ΔG, 12])"},parameterList[[2]][[2]],parameterList[[2]][[3]]},{{n3,n3Z,"(Subscript[ΔG, 21])"},parameterList[[3]][[2]],parameterList[[3]][[3]]},{{n4,n4Z,"(Subscript[ΔG, 22])"},parameterList[[4]][[2]],parameterList[[4]][[3]]},{{s1,s1Z,"(Subscript[ΔS1, 12])"},parameterList[[5]][[2]],parameterList[[5]][[3]]},{{s2,s2Z,"(Subscript[ΔS2, 12])"},parameterList[[6]][[2]],parameterList[[6]][[3]]},{{v1,v1Z,"(Subscript[Δv, 1])"},parameterList[[7]][[2]],parameterList[[7]][[3]]},{{v2,v2Z,"(Subscript[Δv, 2])"},parameterList[[8]][[2]],parameterList[[8]][[3]]},{{v3,v3Z,"(Subscript[Δv, 3])"},parameterList[[9]][[2]],parameterList[[9]][[3]]},{{v4,v4Z,"(Subscript[Δv, 4])"},parameterList[[10]][[2]],parameterList[[10]][[3]]},{{cn1,cn1Z,"(Subscript[cpG, 11])"},parameterList[[11]][[2]],parameterList[[11]][[3]]},{{cn2,cn2Z,"(Subscript[cpG, 12])"},parameterList[[12]][[2]],parameterList[[12]][[3]]},{{cn3,cn3Z,"(Subscript[cpG, 21])"},parameterList[[13]][[2]],parameterList[[13]][[3]]},{{cn4,cn4Z,"(Subscript[cpG, 22])"},parameterList[[14]][[2]],parameterList[[14]][[3]]},{{cs1,s1Z,"(Subscript[cpS1, 12])"},parameterList[[15]][[2]],parameterList[[15]][[3]]},{{cs2,s2Z,"(Subscript[cpS2, 12])"},parameterList[[16]][[2]],parameterList[[16]][[3]]},{{αα,ααZ,"cpBoxSize"},parameterList[[17]][[2]],parameterList[[17]][[3]]},{{ββ,ββZ,"valueBoxSize"},parameterList[[18]][[2]],parameterList[[18]][[3]]},{{inertia,inertiaPD},parameterList[[19]][[2]],parameterList[[19]][[3]]},{{e1,e1z,"(Subscript[e, 1])"},parameterList[[20]][[2]],parameterList[[20]][[3]]},{{e2,e2z,"(Subscript[e, 2])"},parameterList[[21]][[2]],parameterList[[21]][[3]]},{{r,rz,"(Subscript[V, r])"},parameterList[[22]][[2]],parameterList[[22]][[3]]},{{u,uz,"(Subscript[V, u])"},parameterList[[23]][[2]],parameterList[[23]][[3]]},{{σ1,σ1z,"(Subscript[V, (Subscript[s, 1])])"},parameterList[[24]][[2]],parameterList[[24]][[3]]},{{σ2,σ2z,"(Subscript[V, (Subscript[s, 2])])"},parameterList[[25]][[2]],parameterList[[25]][[3]]},{{α,αz,"driven frequency α"},parameterList[[26]][[2]],parameterList[[26]][[3]]},{{β,βz,"driven amplitude β"},parameterList[[27]][[2]],parameterList[[27]][[3]]},{{γ,γz,"gravity amplitude γ"},parameterList[[28]][[2]],parameterList[[28]][[3]]},{{δ,δz,"gravity range δ"},parameterList[[29]][[2]],parameterList[[29]][[3]]},{{tLow,tLowz,"(Subscript[t, Low])"},parameterList[[30]][[2]],parameterList[[30]][[3]]},{{tHigh,tHighz,"(Subscript[t, High])"},parameterList[[31]][[2]],parameterList[[31]][[3]]}]

I think I am missing phase space plots

Manipulate[MatrixForm[{flowNumComplete[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0 ],MatrixForm[combinedPayoff[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0 ]],Chop[Eigenvalues[inversegab[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0 ].combinedPayoff[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0 ]//N]],NullSpace[combinedPayoff[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0 ]],MatrixForm[{{ParametricPlot[{forceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[3]],flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[3]]},{t,tLow,tHigh },AxesLabel->{"(Subscript[f, N])","(Subscript[p, N])"},ColorFunction->Function[{t},Hue[t]],PlotStyle->{Thick,Blue},ImageSize->{300},AspectRatio->1,PlotLegends->{"p1"}],ParametricPlot[{forceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[4]],flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[4]]},{t,tLow,tHigh },AxesLabel->{"(Subscript[f, C])","(Subscript[p, C])"},ColorFunction->Function[{t},Hue[t]],PlotStyle->{Thick,Blue},ImageSize->{300},AspectRatio->1,PlotLegends->{"p1"}]},{ParametricPlot[{forceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[1]],flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[1]]},{t,tLow,tHigh },AxesLabel->{"(Subscript[f, N])","(Subscript[p, N])"},ColorFunction->Function[{t},Hue[t]],PlotStyle->{Thick,Green},ImageSize->{300},AspectRatio->1,PlotLegends->{"p2"}],ParametricPlot[{forceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[2]],flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[2]]},{t,tLow,tHigh },AxesLabel->{"(Subscript[f, C])","(Subscript[p, C])"},ColorFunction->Function[{t},Hue[t]],PlotStyle->{Thick,Green},ImageSize->{300},AspectRatio->1,PlotLegends->{"p2"}]}}]},TableSpacing-> 10,TableAlignments->Left,TableHeadings->{{"Flow Vector","Composite Payoff Matrix", "(g^(-1)).F Eigenvalues","Null Space Vector(s)","Phase Space Plots"},None}],{{n1,n1Z,"(Subscript[ΔG, 11])"},parameterList[[1]][[2]],parameterList[[1]][[3]]},{{n2,n2Z,"(Subscript[ΔG, 12])"},parameterList[[2]][[2]],parameterList[[2]][[3]]},{{n3,n3Z,"(Subscript[ΔG, 21])"},parameterList[[3]][[2]],parameterList[[3]][[3]]},{{n4,n4Z,"(Subscript[ΔG, 22])"},parameterList[[4]][[2]],parameterList[[4]][[3]]},{{s1,s1Z,"(Subscript[ΔS1, 12])"},parameterList[[5]][[2]],parameterList[[5]][[3]]},{{s2,s2Z,"(Subscript[ΔS2, 12])"},parameterList[[6]][[2]],parameterList[[6]][[3]]},{{v1,v1Z,"(Subscript[Δv, 1])"},parameterList[[7]][[2]],parameterList[[7]][[3]]},{{v2,v2Z,"(Subscript[Δv, 2])"},parameterList[[8]][[2]],parameterList[[8]][[3]]},{{v3,v3Z,"(Subscript[Δv, 3])"},parameterList[[9]][[2]],parameterList[[9]][[3]]},{{v4,v4Z,"(Subscript[Δv, 4])"},parameterList[[10]][[2]],parameterList[[10]][[3]]},{{cn1,cn1Z,"(Subscript[cpG, 11])"},parameterList[[11]][[2]],parameterList[[11]][[3]]},{{cn2,cn2Z,"(Subscript[cpG, 12])"},parameterList[[12]][[2]],parameterList[[12]][[3]]},{{cn3,cn3Z,"(Subscript[cpG, 21])"},parameterList[[13]][[2]],parameterList[[13]][[3]]},{{cn4,cn4Z,"(Subscript[cpG, 22])"},parameterList[[14]][[2]],parameterList[[14]][[3]]},{{cs1,s1Z,"(Subscript[cpS1, 12])"},parameterList[[15]][[2]],parameterList[[15]][[3]]},{{cs2,s2Z,"(Subscript[cpS2, 12])"},parameterList[[16]][[2]],parameterList[[16]][[3]]},{{αα,ααZ,"cpBoxSize"},parameterList[[17]][[2]],parameterList[[17]][[3]]},{{ββ,ββZ,"valueBoxSize"},parameterList[[18]][[2]],parameterList[[18]][[3]]},{{inertia,inertiaPD},parameterList[[19]][[2]],parameterList[[19]][[3]]},{{e1,e1z,"(Subscript[e, 1])"},parameterList[[20]][[2]],parameterList[[20]][[3]]},{{e2,e2z,"(Subscript[e, 2])"},parameterList[[21]][[2]],parameterList[[21]][[3]]},{{r,rz,"(Subscript[V, r])"},parameterList[[22]][[2]],parameterList[[22]][[3]]},{{u,uz,"(Subscript[V, u])"},parameterList[[23]][[2]],parameterList[[23]][[3]]},{{σ1,σ1z,"(Subscript[V, (Subscript[s, 1])])"},parameterList[[24]][[2]],parameterList[[24]][[3]]},{{σ2,σ2z,"(Subscript[V, (Subscript[s, 2])])"},parameterList[[25]][[2]],parameterList[[25]][[3]]},{{α,αz,"driven frequency α"},parameterList[[26]][[2]],parameterList[[26]][[3]]},{{β,βz,"driven amplitude β"},parameterList[[27]][[2]],parameterList[[27]][[3]]},{{γ,γz,"gravity amplitude γ"},parameterList[[28]][[2]],parameterList[[28]][[3]]},{{δ,δz,"gravity range δ"},parameterList[[29]][[2]],parameterList[[29]][[3]]},{{tLow,tLowz,"(Subscript[t, Low])"},parameterList[[30]][[2]],parameterList[[30]][[3]]},{{tHigh,tHighz,"(Subscript[t, High])"},parameterList[[31]][[2]],parameterList[[31]][[3]]}]

Based on the solution to the differential equation, the preferences are also determined as a function of time:

Manipulate[MatrixForm[{Plot[Evaluate[preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ]],{t,tLow,tHigh },AxesLabel->{τ,"preference"},PlotStyle->{Green,Purple, Blue,Red,Black},ImageSize->{500},PlotLegends->{"p2:N","p2:C","p1:N","p1:C","time"}],Plot[Evaluate[forceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ]],{t,tLow,tHigh },AxesLabel->{τ,"force"},PlotStyle->{Green,Purple, Blue,Red,Black},ImageSize->{500},PlotLegends->{"p2:(Subscript[f, N])","p2:(Subscript[f, C])","p1:(Subscript[f, N])","p1:(Subscript[f, C])","time"}]}],{{n1,n1Z,"(Subscript[ΔG, 11])"},parameterList[[1]][[2]],parameterList[[1]][[3]]},{{n2,n2Z,"(Subscript[ΔG, 12])"},parameterList[[2]][[2]],parameterList[[2]][[3]]},{{n3,n3Z,"(Subscript[ΔG, 21])"},parameterList[[3]][[2]],parameterList[[3]][[3]]},{{n4,n4Z,"(Subscript[ΔG, 22])"},parameterList[[4]][[2]],parameterList[[4]][[3]]},{{s1,s1Z,"(Subscript[ΔS1, 12])"},parameterList[[5]][[2]],parameterList[[5]][[3]]},{{s2,s2Z,"(Subscript[ΔS2, 12])"},parameterList[[6]][[2]],parameterList[[6]][[3]]},{{v1,v1Z,"(Subscript[Δv, 1])"},parameterList[[7]][[2]],parameterList[[7]][[3]]},{{v2,v2Z,"(Subscript[Δv, 2])"},parameterList[[8]][[2]],parameterList[[8]][[3]]},{{v3,v3Z,"(Subscript[Δv, 3])"},parameterList[[9]][[2]],parameterList[[9]][[3]]},{{v4,v4Z,"(Subscript[Δv, 4])"},parameterList[[10]][[2]],parameterList[[10]][[3]]},{{cn1,cn1Z,"(Subscript[cpG, 11])"},parameterList[[11]][[2]],parameterList[[11]][[3]]},{{cn2,cn2Z,"(Subscript[cpG, 12])"},parameterList[[12]][[2]],parameterList[[12]][[3]]},{{cn3,cn3Z,"(Subscript[cpG, 21])"},parameterList[[13]][[2]],parameterList[[13]][[3]]},{{cn4,cn4Z,"(Subscript[cpG, 22])"},parameterList[[14]][[2]],parameterList[[14]][[3]]},{{cs1,s1Z,"(Subscript[cpS1, 12])"},parameterList[[15]][[2]],parameterList[[15]][[3]]},{{cs2,s2Z,"(Subscript[cpS2, 12])"},parameterList[[16]][[2]],parameterList[[16]][[3]]},{{αα,ααZ,"cpBoxSize"},parameterList[[17]][[2]],parameterList[[17]][[3]]},{{ββ,ββZ,"valueBoxSize"},parameterList[[18]][[2]],parameterList[[18]][[3]]},{{inertia,inertiaPD},parameterList[[19]][[2]],parameterList[[19]][[3]]},{{e1,e1z,"(Subscript[e, 1])"},parameterList[[20]][[2]],parameterList[[20]][[3]]},{{e2,e2z,"(Subscript[e, 2])"},parameterList[[21]][[2]],parameterList[[21]][[3]]},{{r,rz,"(Subscript[V, r])"},parameterList[[22]][[2]],parameterList[[22]][[3]]},{{u,uz,"(Subscript[V, u])"},parameterList[[23]][[2]],parameterList[[23]][[3]]},{{σ1,σ1z,"(Subscript[V, (Subscript[s, 1])])"},parameterList[[24]][[2]],parameterList[[24]][[3]]},{{σ2,σ2z,"(Subscript[V, (Subscript[s, 2])])"},parameterList[[25]][[2]],parameterList[[25]][[3]]},{{α,αz,"driven frequency α"},parameterList[[26]][[2]],parameterList[[26]][[3]]},{{β,βz,"driven amplitude β"},parameterList[[27]][[2]],parameterList[[27]][[3]]},{{γ,γz,"gravity amplitude γ"},parameterList[[28]][[2]],parameterList[[28]][[3]]},{{δ,δz,"gravity range δ"},parameterList[[29]][[2]],parameterList[[29]][[3]]},{{tLow,tLowz,"(Subscript[t, Low])"},parameterList[[30]][[2]],parameterList[[30]][[3]]},{{tHigh,tHighz,"(Subscript[t, High])"},parameterList[[31]][[2]],parameterList[[31]][[3]]}]

Manipulate[ContourPlot[Evaluate[Abs[flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[1]]-flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][tPrime][[1]]]],{t,tLow,tHigh},{tPrime,tLow,tHigh},ColorFunction->Hue,Contours->30,PlotLegends->Automatic,PlotLabel->"player 2:(Subscript[p, N]) (flow) Recurrence Plot"],{{n1,n1Z,"(Subscript[ΔG, 11])"},parameterList[[1]][[2]],parameterList[[1]][[3]]},{{n2,n2Z,"(Subscript[ΔG, 12])"},parameterList[[2]][[2]],parameterList[[2]][[3]]},{{n3,n3Z,"(Subscript[ΔG, 21])"},parameterList[[3]][[2]],parameterList[[3]][[3]]},{{n4,n4Z,"(Subscript[ΔG, 22])"},parameterList[[4]][[2]],parameterList[[4]][[3]]},{{s1,s1Z,"(Subscript[ΔS1, 12])"},parameterList[[5]][[2]],parameterList[[5]][[3]]},{{s2,s2Z,"(Subscript[ΔS2, 12])"},parameterList[[6]][[2]],parameterList[[6]][[3]]},{{v1,v1Z,"(Subscript[Δv, 1])"},parameterList[[7]][[2]],parameterList[[7]][[3]]},{{v2,v2Z,"(Subscript[Δv, 2])"},parameterList[[8]][[2]],parameterList[[8]][[3]]},{{v3,v3Z,"(Subscript[Δv, 3])"},parameterList[[9]][[2]],parameterList[[9]][[3]]},{{v4,v4Z,"(Subscript[Δv, 4])"},parameterList[[10]][[2]],parameterList[[10]][[3]]},{{cn1,cn1Z,"(Subscript[cpG, 11])"},parameterList[[11]][[2]],parameterList[[11]][[3]]},{{cn2,cn2Z,"(Subscript[cpG, 12])"},parameterList[[12]][[2]],parameterList[[12]][[3]]},{{cn3,cn3Z,"(Subscript[cpG, 21])"},parameterList[[13]][[2]],parameterList[[13]][[3]]},{{cn4,cn4Z,"(Subscript[cpG, 22])"},parameterList[[14]][[2]],parameterList[[14]][[3]]},{{cs1,s1Z,"(Subscript[cpS1, 12])"},parameterList[[15]][[2]],parameterList[[15]][[3]]},{{cs2,s2Z,"(Subscript[cpS2, 12])"},parameterList[[16]][[2]],parameterList[[16]][[3]]},{{αα,ααZ,"cpBoxSize"},parameterList[[17]][[2]],parameterList[[17]][[3]]},{{ββ,ββZ,"valueBoxSize"},parameterList[[18]][[2]],parameterList[[18]][[3]]},{{inertia,inertiaPD},parameterList[[19]][[2]],parameterList[[19]][[3]]},{{e1,e1z,"(Subscript[e, 1])"},parameterList[[20]][[2]],parameterList[[20]][[3]]},{{e2,e2z,"(Subscript[e, 2])"},parameterList[[21]][[2]],parameterList[[21]][[3]]},{{r,rz,"(Subscript[V, r])"},parameterList[[22]][[2]],parameterList[[22]][[3]]},{{u,uz,"(Subscript[V, u])"},parameterList[[23]][[2]],parameterList[[23]][[3]]},{{σ1,σ1z,"(Subscript[V, (Subscript[s, 1])])"},parameterList[[24]][[2]],parameterList[[24]][[3]]},{{σ2,σ2z,"(Subscript[V, (Subscript[s, 2])])"},parameterList[[25]][[2]],parameterList[[25]][[3]]},{{α,αz,"driven frequency α"},parameterList[[26]][[2]],parameterList[[26]][[3]]},{{β,βz,"driven amplitude β"},parameterList[[27]][[2]],parameterList[[27]][[3]]},{{γ,γz,"gravity amplitude γ"},parameterList[[28]][[2]],parameterList[[28]][[3]]},{{δ,δz,"gravity range δ"},parameterList[[29]][[2]],parameterList[[29]][[3]]},{{tLow,tLowz,"(Subscript[t, Low])"},parameterList[[30]][[2]],parameterList[[30]][[3]]},{{tHigh,tHighz,"(Subscript[t, High])"},parameterList[[31]][[2]],parameterList[[31]][[3]]}]

Manipulate[ContourPlot[Evaluate[Abs[flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[2]]-flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][tPrime][[2]]]],{t,tLow,tHigh},{tPrime,tLow,tHigh},ColorFunction->Hue,Contours->30,PlotLegends->Automatic,PlotLabel->"player 2:(Subscript[p, C]) (flow) Recurrence Plot"],{{n1,n1Z,"(Subscript[ΔG, 11])"},parameterList[[1]][[2]],parameterList[[1]][[3]]},{{n2,n2Z,"(Subscript[ΔG, 12])"},parameterList[[2]][[2]],parameterList[[2]][[3]]},{{n3,n3Z,"(Subscript[ΔG, 21])"},parameterList[[3]][[2]],parameterList[[3]][[3]]},{{n4,n4Z,"(Subscript[ΔG, 22])"},parameterList[[4]][[2]],parameterList[[4]][[3]]},{{s1,s1Z,"(Subscript[ΔS1, 12])"},parameterList[[5]][[2]],parameterList[[5]][[3]]},{{s2,s2Z,"(Subscript[ΔS2, 12])"},parameterList[[6]][[2]],parameterList[[6]][[3]]},{{v1,v1Z,"(Subscript[Δv, 1])"},parameterList[[7]][[2]],parameterList[[7]][[3]]},{{v2,v2Z,"(Subscript[Δv, 2])"},parameterList[[8]][[2]],parameterList[[8]][[3]]},{{v3,v3Z,"(Subscript[Δv, 3])"},parameterList[[9]][[2]],parameterList[[9]][[3]]},{{v4,v4Z,"(Subscript[Δv, 4])"},parameterList[[10]][[2]],parameterList[[10]][[3]]},{{cn1,cn1Z,"(Subscript[cpG, 11])"},parameterList[[11]][[2]],parameterList[[11]][[3]]},{{cn2,cn2Z,"(Subscript[cpG, 12])"},parameterList[[12]][[2]],parameterList[[12]][[3]]},{{cn3,cn3Z,"(Subscript[cpG, 21])"},parameterList[[13]][[2]],parameterList[[13]][[3]]},{{cn4,cn4Z,"(Subscript[cpG, 22])"},parameterList[[14]][[2]],parameterList[[14]][[3]]},{{cs1,s1Z,"(Subscript[cpS1, 12])"},parameterList[[15]][[2]],parameterList[[15]][[3]]},{{cs2,s2Z,"(Subscript[cpS2, 12])"},parameterList[[16]][[2]],parameterList[[16]][[3]]},{{αα,ααZ,"cpBoxSize"},parameterList[[17]][[2]],parameterList[[17]][[3]]},{{ββ,ββZ,"valueBoxSize"},parameterList[[18]][[2]],parameterList[[18]][[3]]},{{inertia,inertiaPD},parameterList[[19]][[2]],parameterList[[19]][[3]]},{{e1,e1z,"(Subscript[e, 1])"},parameterList[[20]][[2]],parameterList[[20]][[3]]},{{e2,e2z,"(Subscript[e, 2])"},parameterList[[21]][[2]],parameterList[[21]][[3]]},{{r,rz,"(Subscript[V, r])"},parameterList[[22]][[2]],parameterList[[22]][[3]]},{{u,uz,"(Subscript[V, u])"},parameterList[[23]][[2]],parameterList[[23]][[3]]},{{σ1,σ1z,"(Subscript[V, (Subscript[s, 1])])"},parameterList[[24]][[2]],parameterList[[24]][[3]]},{{σ2,σ2z,"(Subscript[V, (Subscript[s, 2])])"},parameterList[[25]][[2]],parameterList[[25]][[3]]},{{α,αz,"driven frequency α"},parameterList[[26]][[2]],parameterList[[26]][[3]]},{{β,βz,"driven amplitude β"},parameterList[[27]][[2]],parameterList[[27]][[3]]},{{γ,γz,"gravity amplitude γ"},parameterList[[28]][[2]],parameterList[[28]][[3]]},{{δ,δz,"gravity range δ"},parameterList[[29]][[2]],parameterList[[29]][[3]]},{{tLow,tLowz,"(Subscript[t, Low])"},parameterList[[30]][[2]],parameterList[[30]][[3]]},{{tHigh,tHighz,"(Subscript[t, High])"},parameterList[[31]][[2]],parameterList[[31]][[3]]}]

Manipulate[ContourPlot[Evaluate[Abs[flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[3]]-flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][tPrime][[3]]]],{t,tLow,tHigh},{tPrime,tLow,tHigh},ColorFunction->Hue,Contours->30,PlotLegends->Automatic,PlotLabel->"player 1:(Subscript[p, N]) (flow) Recurrence Plot"],{{n1,n1Z,"(Subscript[ΔG, 11])"},parameterList[[1]][[2]],parameterList[[1]][[3]]},{{n2,n2Z,"(Subscript[ΔG, 12])"},parameterList[[2]][[2]],parameterList[[2]][[3]]},{{n3,n3Z,"(Subscript[ΔG, 21])"},parameterList[[3]][[2]],parameterList[[3]][[3]]},{{n4,n4Z,"(Subscript[ΔG, 22])"},parameterList[[4]][[2]],parameterList[[4]][[3]]},{{s1,s1Z,"(Subscript[ΔS1, 12])"},parameterList[[5]][[2]],parameterList[[5]][[3]]},{{s2,s2Z,"(Subscript[ΔS2, 12])"},parameterList[[6]][[2]],parameterList[[6]][[3]]},{{v1,v1Z,"(Subscript[Δv, 1])"},parameterList[[7]][[2]],parameterList[[7]][[3]]},{{v2,v2Z,"(Subscript[Δv, 2])"},parameterList[[8]][[2]],parameterList[[8]][[3]]},{{v3,v3Z,"(Subscript[Δv, 3])"},parameterList[[9]][[2]],parameterList[[9]][[3]]},{{v4,v4Z,"(Subscript[Δv, 4])"},parameterList[[10]][[2]],parameterList[[10]][[3]]},{{cn1,cn1Z,"(Subscript[cpG, 11])"},parameterList[[11]][[2]],parameterList[[11]][[3]]},{{cn2,cn2Z,"(Subscript[cpG, 12])"},parameterList[[12]][[2]],parameterList[[12]][[3]]},{{cn3,cn3Z,"(Subscript[cpG, 21])"},parameterList[[13]][[2]],parameterList[[13]][[3]]},{{cn4,cn4Z,"(Subscript[cpG, 22])"},parameterList[[14]][[2]],parameterList[[14]][[3]]},{{cs1,s1Z,"(Subscript[cpS1, 12])"},parameterList[[15]][[2]],parameterList[[15]][[3]]},{{cs2,s2Z,"(Subscript[cpS2, 12])"},parameterList[[16]][[2]],parameterList[[16]][[3]]},{{αα,ααZ,"cpBoxSize"},parameterList[[17]][[2]],parameterList[[17]][[3]]},{{ββ,ββZ,"valueBoxSize"},parameterList[[18]][[2]],parameterList[[18]][[3]]},{{inertia,inertiaPD},parameterList[[19]][[2]],parameterList[[19]][[3]]},{{e1,e1z,"(Subscript[e, 1])"},parameterList[[20]][[2]],parameterList[[20]][[3]]},{{e2,e2z,"(Subscript[e, 2])"},parameterList[[21]][[2]],parameterList[[21]][[3]]},{{r,rz,"(Subscript[V, r])"},parameterList[[22]][[2]],parameterList[[22]][[3]]},{{u,uz,"(Subscript[V, u])"},parameterList[[23]][[2]],parameterList[[23]][[3]]},{{σ1,σ1z,"(Subscript[V, (Subscript[s, 1])])"},parameterList[[24]][[2]],parameterList[[24]][[3]]},{{σ2,σ2z,"(Subscript[V, (Subscript[s, 2])])"},parameterList[[25]][[2]],parameterList[[25]][[3]]},{{α,αz,"driven frequency α"},parameterList[[26]][[2]],parameterList[[26]][[3]]},{{β,βz,"driven amplitude β"},parameterList[[27]][[2]],parameterList[[27]][[3]]},{{γ,γz,"gravity amplitude γ"},parameterList[[28]][[2]],parameterList[[28]][[3]]},{{δ,δz,"gravity range δ"},parameterList[[29]][[2]],parameterList[[29]][[3]]},{{tLow,tLowz,"(Subscript[t, Low])"},parameterList[[30]][[2]],parameterList[[30]][[3]]},{{tHigh,tHighz,"(Subscript[t, High])"},parameterList[[31]][[2]],parameterList[[31]][[3]]}]

Manipulate[ContourPlot[Evaluate[Abs[flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[4]]-flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][tPrime][[4]]]],{t,tLow,tHigh},{tPrime,tLow,tHigh},ColorFunction->Hue,Contours->30,PlotLegends->Automatic,PlotLabel->"player 1:(Subscript[p, C]) (flow) Recurrence Plot"],{{n1,n1Z,"(Subscript[ΔG, 11])"},parameterList[[1]][[2]],parameterList[[1]][[3]]},{{n2,n2Z,"(Subscript[ΔG, 12])"},parameterList[[2]][[2]],parameterList[[2]][[3]]},{{n3,n3Z,"(Subscript[ΔG, 21])"},parameterList[[3]][[2]],parameterList[[3]][[3]]},{{n4,n4Z,"(Subscript[ΔG, 22])"},parameterList[[4]][[2]],parameterList[[4]][[3]]},{{s1,s1Z,"(Subscript[ΔS1, 12])"},parameterList[[5]][[2]],parameterList[[5]][[3]]},{{s2,s2Z,"(Subscript[ΔS2, 12])"},parameterList[[6]][[2]],parameterList[[6]][[3]]},{{v1,v1Z,"(Subscript[Δv, 1])"},parameterList[[7]][[2]],parameterList[[7]][[3]]},{{v2,v2Z,"(Subscript[Δv, 2])"},parameterList[[8]][[2]],parameterList[[8]][[3]]},{{v3,v3Z,"(Subscript[Δv, 3])"},parameterList[[9]][[2]],parameterList[[9]][[3]]},{{v4,v4Z,"(Subscript[Δv, 4])"},parameterList[[10]][[2]],parameterList[[10]][[3]]},{{cn1,cn1Z,"(Subscript[cpG, 11])"},parameterList[[11]][[2]],parameterList[[11]][[3]]},{{cn2,cn2Z,"(Subscript[cpG, 12])"},parameterList[[12]][[2]],parameterList[[12]][[3]]},{{cn3,cn3Z,"(Subscript[cpG, 21])"},parameterList[[13]][[2]],parameterList[[13]][[3]]},{{cn4,cn4Z,"(Subscript[cpG, 22])"},parameterList[[14]][[2]],parameterList[[14]][[3]]},{{cs1,s1Z,"(Subscript[cpS1, 12])"},parameterList[[15]][[2]],parameterList[[15]][[3]]},{{cs2,s2Z,"(Subscript[cpS2, 12])"},parameterList[[16]][[2]],parameterList[[16]][[3]]},{{αα,ααZ,"cpBoxSize"},parameterList[[17]][[2]],parameterList[[17]][[3]]},{{ββ,ββZ,"valueBoxSize"},parameterList[[18]][[2]],parameterList[[18]][[3]]},{{inertia,inertiaPD},parameterList[[19]][[2]],parameterList[[19]][[3]]},{{e1,e1z,"(Subscript[e, 1])"},parameterList[[20]][[2]],parameterList[[20]][[3]]},{{e2,e2z,"(Subscript[e, 2])"},parameterList[[21]][[2]],parameterList[[21]][[3]]},{{r,rz,"(Subscript[V, r])"},parameterList[[22]][[2]],parameterList[[22]][[3]]},{{u,uz,"(Subscript[V, u])"},parameterList[[23]][[2]],parameterList[[23]][[3]]},{{σ1,σ1z,"(Subscript[V, (Subscript[s, 1])])"},parameterList[[24]][[2]],parameterList[[24]][[3]]},{{σ2,σ2z,"(Subscript[V, (Subscript[s, 2])])"},parameterList[[25]][[2]],parameterList[[25]][[3]]},{{α,αz,"driven frequency α"},parameterList[[26]][[2]],parameterList[[26]][[3]]},{{β,βz,"driven amplitude β"},parameterList[[27]][[2]],parameterList[[27]][[3]]},{{γ,γz,"gravity amplitude γ"},parameterList[[28]][[2]],parameterList[[28]][[3]]},{{δ,δz,"gravity range δ"},parameterList[[29]][[2]],parameterList[[29]][[3]]},{{tLow,tLowz,"(Subscript[t, Low])"},parameterList[[30]][[2]],parameterList[[30]][[3]]},{{tHigh,tHighz,"(Subscript[t, High])"},parameterList[[31]][[2]],parameterList[[31]][[3]]}]

Manipulate[ContourPlot[Evaluate[Abs[flowNumComplete[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[5]]-flowNumComplete[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][tPrime][[5]]]],{t,tLow,tHigh},{tPrime,tLow,tHigh},ColorFunction->Hue,Contours->30,PlotLegends->Automatic,PlotLabel->"time flow (Subscript[p, t]) Recurrence Plot"],{{n1,n1Z,"(Subscript[ΔG, 11])"},parameterList[[1]][[2]],parameterList[[1]][[3]]},{{n2,n2Z,"(Subscript[ΔG, 12])"},parameterList[[2]][[2]],parameterList[[2]][[3]]},{{n3,n3Z,"(Subscript[ΔG, 21])"},parameterList[[3]][[2]],parameterList[[3]][[3]]},{{n4,n4Z,"(Subscript[ΔG, 22])"},parameterList[[4]][[2]],parameterList[[4]][[3]]},{{s1,s1Z,"(Subscript[ΔS1, 12])"},parameterList[[5]][[2]],parameterList[[5]][[3]]},{{s2,s2Z,"(Subscript[ΔS2, 12])"},parameterList[[6]][[2]],parameterList[[6]][[3]]},{{v1,v1Z,"(Subscript[Δv, 1])"},parameterList[[7]][[2]],parameterList[[7]][[3]]},{{v2,v2Z,"(Subscript[Δv, 2])"},parameterList[[8]][[2]],parameterList[[8]][[3]]},{{v3,v3Z,"(Subscript[Δv, 3])"},parameterList[[9]][[2]],parameterList[[9]][[3]]},{{v4,v4Z,"(Subscript[Δv, 4])"},parameterList[[10]][[2]],parameterList[[10]][[3]]},{{cn1,cn1Z,"(Subscript[cpG, 11])"},parameterList[[11]][[2]],parameterList[[11]][[3]]},{{cn2,cn2Z,"(Subscript[cpG, 12])"},parameterList[[12]][[2]],parameterList[[12]][[3]]},{{cn3,cn3Z,"(Subscript[cpG, 21])"},parameterList[[13]][[2]],parameterList[[13]][[3]]},{{cn4,cn4Z,"(Subscript[cpG, 22])"},parameterList[[14]][[2]],parameterList[[14]][[3]]},{{cs1,s1Z,"(Subscript[cpS1, 12])"},parameterList[[15]][[2]],parameterList[[15]][[3]]},{{cs2,s2Z,"(Subscript[cpS2, 12])"},parameterList[[16]][[2]],parameterList[[16]][[3]]},{{αα,ααZ,"cpBoxSize"},parameterList[[17]][[2]],parameterList[[17]][[3]]},{{ββ,ββZ,"valueBoxSize"},parameterList[[18]][[2]],parameterList[[18]][[3]]},{{inertia,inertiaPD},parameterList[[19]][[2]],parameterList[[19]][[3]]},{{e1,e1z,"(Subscript[e, 1])"},parameterList[[20]][[2]],parameterList[[20]][[3]]},{{e2,e2z,"(Subscript[e, 2])"},parameterList[[21]][[2]],parameterList[[21]][[3]]},{{r,rz,"(Subscript[V, r])"},parameterList[[22]][[2]],parameterList[[22]][[3]]},{{u,uz,"(Subscript[V, u])"},parameterList[[23]][[2]],parameterList[[23]][[3]]},{{σ1,σ1z,"(Subscript[V, (Subscript[s, 1])])"},parameterList[[24]][[2]],parameterList[[24]][[3]]},{{σ2,σ2z,"(Subscript[V, (Subscript[s, 2])])"},parameterList[[25]][[2]],parameterList[[25]][[3]]},{{α,αz,"driven frequency α"},parameterList[[26]][[2]],parameterList[[26]][[3]]},{{β,βz,"driven amplitude β"},parameterList[[27]][[2]],parameterList[[27]][[3]]},{{γ,γz,"gravity amplitude γ"},parameterList[[28]][[2]],parameterList[[28]][[3]]},{{δ,δz,"gravity range δ"},parameterList[[29]][[2]],parameterList[[29]][[3]]},{{tLow,tLowz,"(Subscript[t, Low])"},parameterList[[30]][[2]],parameterList[[30]][[3]]},{{tHigh,tHighz,"(Subscript[t, High])"},parameterList[[31]][[2]],parameterList[[31]][[3]]}]

Summary Format Numerical results including recurrence graphs with Sensitivity Analysis

The initial payoff matrix is set by the various parameters, as are the eigenvalues and null space vector(s). With the corresponding plot of the time dependence of the strategies, we can see how the eigenvalues are associated with harmonic and exponential behaviors.

Manipulate[MatrixForm[{flowNumComplete[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0 ],MatrixForm[combinedPayoff[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0 ]],Chop[Eigenvalues[inversegab[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0 ].combinedPayoff[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0 ]//N]],NullSpace[combinedPayoff[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0 ]],MatrixForm[{{ParametricPlot[{forceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[3]],flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[3]]},{t,tLow,tHigh },AxesLabel->{"(Subscript[f, N])","(Subscript[p, N])"},ColorFunction->Function[{t},Hue[t]],PlotStyle->{Thick,Blue},ImageSize->{graphSize},AspectRatio->1,PlotLegends->{"p1"}],ParametricPlot[{forceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[4]],flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[4]]},{t,tLow,tHigh },AxesLabel->{"(Subscript[f, C])","(Subscript[p, C])"},ColorFunction->Function[{t},Hue[t]],PlotStyle->{Thick,Blue},ImageSize->{graphSize},AspectRatio->1,PlotLegends->{"p1"}]},{ParametricPlot[{forceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[1]],flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[1]]},{t,tLow,tHigh },AxesLabel->{"(Subscript[f, N])","(Subscript[p, 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Recurrence Plot",ImageSize->{graphSize}],ContourPlot[Evaluate[Abs[flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[4]]-flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][tPrime][[4]]]],{t,tLow,tHigh},{tPrime,tLow,tHigh},ColorFunction->Hue,Contours->30,PlotLegends->Automatic,PlotLabel->"player 1:(Subscript[p, C]) Recurrence Plot",ImageSize->{graphSize}]},{ContourPlot[Evaluate[Abs[flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[1]]-flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][tPrime][[1]]]],{t,tLow,tHigh},{tPrime,tLow,tHigh},ColorFunction->Hue,Contours->30,PlotLegends->Automatic,PlotLabel->"player 2:(Subscript[p, N]) Recurrence Plot",ImageSize->{graphSize}],ContourPlot[Evaluate[Abs[flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[2]]-flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][tPrime][[2]]]],{t,tLow,tHigh},{tPrime,tLow,tHigh},ColorFunction->Hue,Contours->30,PlotLegends->Automatic,PlotLabel->"player 2:(Subscript[p, C]) Recurrence Plot",ImageSize->{graphSize}]},{ContourPlot[Evaluate[Abs[flowNumComplete[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[5]]-flowNumComplete[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][tPrime][[5]]]],{t,tLow,tHigh},{tPrime,tLow,tHigh},ColorFunction->Hue,Contours->30,PlotLegends->Automatic,PlotLabel->"Flow time Recurrence 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Fixed Frame Model--complete solutions

Conclusions

Non-linear effects generate chaotic behaviors

-These show up as structures in recurrence plots and phase space plots

-physics example of a pendulum
-existence of non-linear mechanisms evident already in rock scissors paper
-Used simple model to demonstrate
-More elaborate model confirmed

Effects remain in full theory

-Full theory enforces more consistency between feedback loops
-System dynamics argues the importance of taking into account all feedback loops
-Even seen in process analysis (CMM model for software development, accreditation models for universities, etc.)

Decision making process is deterministic, elastic, non-stochastic, can be chaotic

-As in physics, non-linear effects can make predictions difficult because of sensitivity of predictions to initial conditions
-Predictions are sensitive to the connectivity (elasticity): energy can move between subsystems
-Systems dynamics results: Hit (driver amplitude) what looks like a stable system and you may get chaotic phenomena
-See this experimentally, such as in stock market example