Spatial distribution of chaotic orbits, Gauss Iterated Map and Ergodic Theorems

Chaos theory is a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions. This project focuses on the chaotic nature and spatial distribution of the Gauss iterated map - a nonlinear iterated map given by the Gaussian function. In addition, the Ergodic theorem is used to discuss the relationship between time average and space average, as well as how it is related to finding the Lyapunov exponent using the invariant density function.

What is a Chaotic Map or Function?

A function or map f is chaotic on some interval I if it has the following 3 properties:

1. f has SDIC (Sensitive Dependence on Initial Conditions) on I or f has a positive Lyapunov exponent at each point in I that is not eventually periodic
2. f has a dense set of periodic points on I
3. f is transitive on I or there exists a point x in I whose orbit is dense on I

Why do we want to look at the SPATIAL DISTRIBUTION of the chaotic orbits?

Suppose we have 2 different chaotic maps f and g which both map on the same interval. Given the same seed the chaotic orbits of the 2 functions will follow some random patterns given both functions are chaotic. So how do we distinguish between different chaotic orbits? One of the answers is to look at how the iterates are distributed over I.

Example 1 Spatial distribution of visitation frequencies

Even though both the shifted tent map with a=0.25 and the logistic map with a=4 have indistinguishable chaotic orbits, their spatial distribution of visitation frequencies look very different as shown in the following graphs.

Gauss Iterated Map

The Gauss iterated map (also known as Gaussian map or mouse map) is a non-linear one-dimensional iterated map that maps the bell-shaped Gaussian function into a finite real interval. In the past, the Gauss iterated map has drawn less attention due to its similarity to the logistic map which was studied significantly by people. It has 2 parameters rather than 1 compared to the logistic map and in some cases, there is co-existence of attractors and reverse period-doubling which are also not seen in the logistic map.

The Gauss iterated map is also known as the Mouse map because its bifurcation diagram looks like a mouse.
Wikipedia's Page on Gauss iterated map

Characteristics of the Gauss Iterated Map investigated

• Periodic points of F(x)
• n-cycle AND period-n point of F(x) [Sharkovskii's Theorem]
• Lyapunov exponents of F(x)
• Sensitive Dependence on Initial Conditions of F(x)
• Invariant (probability) density function that defines the invariant measure associated with F(x)

Example 2 Lyapunov exponents of the Gauss Iterated Map

Analysis of the spatial distribution

• There is never a uniform portion of the distribution like the distribution of the shifted tent map (this can be explained by the Law of Conservation of Iterates)
• Self-similarity features (i.e. it contains several copies which are approximately similar to itself)

Example 3 Spatial distribution of the Gauss Iterated Map

Ergodic Theorem and the Invariant Density Function

• Time Average = Space Average
• Using Ergodic Theorem to find the Lyapunov exponent analytically using the invariant density function

Key results of the Gauss iterated map on general parameter space (b, c)

1. The Gauss iterated map can be chaotic depends on its parameters b and c as well as the initial condition
2. Co-existence of 2 attractors (where a fixed point co-exists with some other attractor (chaotic, fixed point, period-2, period-3)) only occurs when the Gauss iterated map has 3 fixed points (however the converse is not true, existence of 3 fixed points do not guarantee the co-existence of 2 attractors)
3. Within the co-existence of 2 attractors parameter, a fixed point and a chaotic attractor co-exist for some parameter space
4. To which attractor does the iterates get attracted to depend on the choice of the seed (therefore the Gauss iterated map can be very sensitive to initial conditions which is an ingredient of chaotic behavior)

Closing Thoughts

In general chaos theory can be very mathematically involved but also has many useful applications in various fields such as physics. Also, many of the commonly known maps such as logistic map has many important connections with real-world applications. This project is a combination of mathematical analysis and computer programming, and all programming are done using Matlab and R.

Last updated on Jan 1, 2019