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Analyzing Stability in Dynamical Systems

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Analyzing Stability in Dynamical Systems

Jade Holst, Wartburg College

Daniel Harrison, Asbury College

Mohamed Jardak, Florida State University

The logistic equation is a classic example of a potentially chaotic dynamical system. We have

Figure1 (top) demonstrates the chaotic nature of the logistic system by plotting its asymptotic solutions for a range of values of the parameter r. This is called a bifurcation map .

Another commonly used method for analyzing stability of dynamical systems in one or several dimensions is the lyapunov exponent (Figure1 bottom.). These exponents measure the rate of divergence of orbits originating from arbitrarily close initial conditions. That is, they measure a system’s sensitivity to its initial conditions (Note the correlation between the lyapunov exponent and the bifurcation map). A positive lyapunov exponent indicates that the system is chaotic. The exponents are described as

n is the number of iteration of the dynamical system and x0is the initial condition.

Figure1

As a stochastic counterpart, we used a variation of the numerical approximation for the

maximum lyapunov exponent. Precisely, the derivative is replaced by a difference quotient,

, where is a random variable. Figure2 is a comparison of

the means obtained by using this counterpart and different values of alpha.

Figure2

But How do we Analyze the Stability in Higher Dimensions?

Lorenz Equations

Initial Conditions = (-1.356, -2.492152,

12.31741)

Parameters: σ = 10, ρ = 28, β = 8/3

ΔT* = .01

Figure 3

Solver: Runge-Kutta (4,5) Built in solver in Matlab

Rossler Equations

Initial Conditions = (-2.209787, 1353531,

0.070299)

Parameters: a = 0.2, b = 0.2, c = 5.7

ΔT* = .01

Figure 4

Solver: Runge-Kutta (4,5) Built in solver in Matlab

Van der Pol

Initial Conditions = (0.1, 0.1, 0.1)

Parameters: b = .01, c = 1

ΔT* = .01

Figure 5

Solver: Runge-Kutta (4,5) Built in solver in Matlab

We look at One Dimension at a time!

A poincare section is often used to reduce a three dimensional (or higher) continuous system to a descrete map of dimension one or two. The strength behind this tool is that these sections have the same topological properties as their continuous counterparts. Often, the local maxima of a variable are used as a one dimensional poincare map (This is easily seen in Figure6.). For more complicated systems, the distance between the maxima is more descriptive of the system’s characteristics. Thus, bifurcation diagrams that are similar to that of the logistic equation can be generated by plotting points of the section after many iterations of the dynamical system (Figure7). Once again, this procedure is applied to a range of values of a given parameter. a. refers to the Rossler system, b. to the Van der Pol, and c. to the Lorenz.

These figures contain plots of the lyapunov exponents (following the Y axis) for the three systems in question. Note that the Van der Pol lyapunov exponent does not correlate with corresponding bifurcation map. This remains a mystery for us.

These graphics display the variance and mean solutions to the Lorenz, Rossler, and Van der Pol equations under the stochastic process described previously, that is, using a Monte Carlo Simulation.

Cell-to-cell mapping is another tool we are researching. This method plots periodic orbits that stay within a given surface. Thus, it delineates the attractors in the surface. The stochastic counterpart may involve replacing one of the parameters with a stochastic process.