Visualizing behaviors in differential geometries using Mathematica

Gerald H. Thomas

MSOE

Abstract

Visualizing behaviors in differential geometries using Mathematica: Differential geometries use partial differential equations to generate the geodesic behaviors that characterize the problem space. Examples are physical models such as Einstein’s theory of relativity, Newton’s classical theory of mechanics or Maxwell’s theory of electricity and magnetism. My interest has been to investigate the possibility of using differential geometry in the social sciences; in particular to gain insight into the causal behavior of decision processes. The starting point is the conjecture that decisions are events that form a continuous manifold in both time (causality) and space (strategies). This is a fundamentally different starting point from the usual stochastic approaches used in the social sciences. 

The challenge is to visualize the behaviors that are consequences of the differential geometry. This talk addresses that challenge. We apply Mathematica solutions to the partial differential equations and look at limit cycles and limit surfaces and their relationship to harmonic solutions and chaotic behaviors. We proceed with model solutions using Mathematica of more complicated differential geometry possibilities that have applications to gravitational theories and decision theories. We find that the NDSolve and Manipulate functionality can be used to great advantage in order to visualize the behaviors in these geometries. There are many other applications of these visualizations, such as teaching possibilities in Electro-Magnetism courses.

Outline

. Decisions as Geometry
  . Take an example www (work-wealth-wisdom) model
  . What is time? What are space components? How is uncertainty dealt with?
  . Idiosyncratic behavior and symmetry
. Steady state solutions
  . Can steady state be chaotic? Jupiter red spot!
  . Examples from WWW--rotations, Coriolis, magnetic fields and payoffs
  . Need to solve complex stationary problem
  . Use harmonic series approximation
  . Do we exclude chaotic behaviors? See toy model . Build on this with harmonic solutions
. Decision making is more complicated than Newtonian physics
  . Two different β transformations to stationary frame--two versions of time
. Communication travels at a finte speed
  . There are communication ellipses

Decisions as Geometry

Position and uncertainty

Game theory concept of mixed strategy translates into the concept of (strategic) position
Game theory is a static theory; this theory is a dynamic extension
Keep to original idea that decisions are deterministic, but can be made with a frequency set by personal preference
The frequency follows a deterministic path, not the individual decision

Time

Sample time intervals have to be long enough for the strategic frequency to be quantified
With this caveat, dynamic behavior evolves in a continuous fashion
Coordinate definition of time--potential surface

Symmetry and idiosyncratic behavior

Decisions are inherently idiosyncratic--we associate such behavior with a symmetry for each player/agent in the decision process
We also associate a symmetry (and a conservation law) to each code of conduct
The symmetries are inactive strategies
What remains are competitive behaviors between the active strategies
The importance of inactive strategies and their interplay with active strategies is something we think is new

WWW model

Work, Wealth, Wisdom (i.e. decisions) Model is a simple but realistic example. Two players: One that focuses on Work, the other on Wealth. 
There are 10:1 Workers to Wealthy. 
The strategies open to Work are “earn” salary or steal. 
The strategies open to Wealthy are use wealth to invest in capital to grow society, or use wealth for rent. 
The space consists of two active strategies for each player, one inactive strategy for each player and time.
Geometry is 7 dimensions=6+1
To each player, there is a payoff field (Coriolis or Magnetic Field force), a scalar field (centripetal force) and a conserved charge
The model is stationary in that there is a frame in which time is inactive: flows are stationary.
In a “normal” frame of reference we say the time dependence is steady-state (standing wave behavior)

Steady State Solutions

Can steady state be chaotic?

Fluid Mechanics: Jupiter, Hurricanes

Note Coriolis, Magnetism, and Payoff behaviors are deeply connected

Solve the complicated partial differential equations without losing possibility for chaotic behaviors

Harmonic series and spectral decomposition approximations proposed

Resultant payoff behavior is complex: for example vector potential has complex gauge condition

A Steady State can be chaotic

Consider a toy model first, Laplace equation with a sin(g) term, much like a physical pendulum

cycles=5
mesh = 16
ω=0
init=1.7
solutiong[z_]=First[NDSolve[{g''[z]+(ω^2)g[z]+ Sin[ g[z]]==init Sum[(( Sin[m z])/m),{m,1,mesh}],g'[-Pi]==g'[Pi],g[Pi]==g[-Pi]},{g[z],g'[z]},{z,-cycles Pi,cycles Pi},MaxSteps->∞]]
gNum[z_]=g[z]/.solutiong[z]
gNumPrime[z_]=g'[z]/.solutiong[z]
Plot[Sum[(( Sin[m z])/m),{m,1,mesh}],{z,- cycles Pi,cycles Pi},PlotStyle->Green,PlotLabel->"Driver"]
Plot[{gNum[z],gNumPrime[z]},{z,-cycles Pi,cycles Pi},PlotStyle->{Blue,Red},PlotLabel->"n cycles of NDSolve g[z] (blue) and g'[z] (red)"]
ParametricPlot[{gNum[z],gNumPrime[z]},{z,-cycles Pi,cycles Pi},AspectRatio->1,AxesLabel->{"g[z]","g'[z]"},PlotStyle->Purple]

Use Manipulate to make chaotic behavior visible

Clear[init]
solution=ParametricNDSolve[{g''[z]+(ω^2)g[z]+ Sin[ g[z]]==init Sum[(( Sin[m z])/m),{m,1,mesh}],g'[-Pi]==g'[Pi],g[Pi]==g[-Pi]},{g},{z,-cycles Pi,cycles Pi},{{init,0,3}}]
gN[init_][z_]=g[init][z]/.solution
gPrimeN[init_][z_]=g[init]'[z]/.solution
Manipulate[ParametricPlot[{gN[init][z],gPrimeN[init][z]},{z,-cycles Pi,cycles Pi},AspectRatio->1,AxesLabel->{"g[z]","g'[z]"},PlotStyle->Purple],{{init,1.5},0,2}]

Compare time-integration (method of lines) result with harmonic series approximation

We see that the harmonnic solution captures some of the chaotic behavior but is itself a cycle and not chaotic. Different starting points however capture different parts of the chaotic  behavior, so in that sense one can still explore chaos

Decision Geometry is more complicated than Newtonian Physics

Which way does time flow? Orthogonal to space, but which way is that? 
Along the time axis as defined as normal to the surface of constant time, or in the direction normal to the spacial hypersurface dx^dy^dz...? 
There are geometries in which they are not equal! In particular the WWW Geometry
We need to visualize each case: look at the velocity along each of these direction possibilities, here called the now-β

Transformation to rest frame: now-β defined by coordinate surface normal (spectral decomposition)

(Subscript[β, υ])=((Subscript[E, υ]^t)/((Subscript[E, ο]^t)(Sqrt[1+(e^α)(Subscript[e, α])])))
x=rcosθ
y=rsinθ

z=0.05, r=1/5

z=-0.20, r=1/5

Transformation to rest frame: dual now-*β defined by surface vectors

*(β^υ)=((Subscript[E, t]^υ)/((Subscript[E, t]^ο)(Sqrt[1+(e^α)(Subscript[e, α])])))

z=0.05, r=1/5

z=-0.20, r=1/5

(generalized) Cross product of surface vectors not equal to surface normal!!

Communication between events has a finite speed

The maximum speed of information in decision geometry is set by the null vector ((dτ)/dt)=0. This is the analog of the speed of light in physics. The resultant equation defines the “light” cone.
Real decisions occur when the shape is an ellipsoid.
The velocity of “light” can never be zero or infinite. Thus a hyperboloid which is mathematically possible is not acceptable for real decisions.
This is important to visualize as in the WWW geometry, the coefficients may change with position.

(Subscript[g, mn]) ((du^m)/dt)((du^n)/dt)+2(Subscript[g, tm]) ((du^m)/dt)+(Subscript[g, tt])=0

Communication ellipsoids/hyperboloids

Manipulate is very useful to see the behavior change as a function of position, but not illustrated here 
Not only is the velocity limited to be between zero and infinity, so also is the momentum, which is the dual vector.
The constraint on the velocity is not sufficient to prove that the constraint on momentum is satisfied, and vice versa.

(Subscript[π, m])=(Subscript[g, mn])((du^n)/dt)
(g^mn) (Subscript[π, m])(Subscript[π, n])+2(g^tm) (Subscript[π, m])+(g^tt)=0