Analyzing Stability in Dynamical Systems Jade Holst, Wartburg College Daniel Harrison, Asbury College Mohamed Jardak, Florida State University

One Dimensional Maps 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 x0 is the initial condition. Figure1

One Dimensional Maps (cont.) 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

Higher Dimensional Maps 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

Higher Dimensional Maps (cont.) 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

Higher Dimensional Maps (cont.) 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

Higher Dimensional Maps (cont.) We look at One Dimension at a time!

Higher Dimensional Maps (cont.) 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.

Higher Dimensional Maps (cont.) 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.

Stochastic Initial Conditions

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.

What’s next? 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.