1. Polynomial Chaos For Dynamical Systems Anatoly Zlotnik, Case Western Reserve University Mohamed Jardak, Florida State University

2. The goal of this project is to implement a polynomial chaos representation to characterize the behavior of random dynamical systems. Polynomial chaos, or PC, can provide information about the probability density of the solution to a system of ordinary differential equations with random components as an alternative to costly Monte Carlo methods. The project includes examination of numerical aspects of PC implementation. A fast unified algorithm has been developed for calculating the necessary coefficients using an analytic technique. Rapid computation of the necessary data now allows for direct calculation for systems with higher order nonlinear terms, leading to a re-evaluation of accuracy in pseudo-spectral, or PS, approximations. This will also lead to higher numerical stability for applications to highly nonlinear systems. Abstract

3. Stochastic Differential Equations and Spectral Collocation Method

4. Stochastic Differential Equations and Spectral Collocation Method (cont.)

5. Solutions are obtained using an 8th degree PC expansion and the ode45 routine in MATLAB with relative integration tolerance at 1e-6. P+1 ODEs are solved, giving a stochastic series expansion at all grid points. The series is used at each point to simulate a data set by Monte Carlo, and to estimate a PDF. Color plots are used to illustrate the variation of the solution PDF with time. Blue and red indicate low and high probability density, respectively. Applications To Dynamical Systems

6. Applications To Dynamical Systems (cont.)

8. Lorenz System Initial conditions in x, y, and z are normal random with mean=0.01 and std=0.001. The coefficients sigma, rho, and beta are the standard 10, 28, and 8/3, respectively. The spatial variables are expanded in PC. As the system moves around the first attractor, the solution PDF has a small range and high probability density. At T=23, The position of the oscillator becomes highly uncertain. The PDF has a much broader range, and lower density. The profiles of the PDF at various times also display this behavior. The solution undergoes a sort of “bifurcation” at certain times, where it is likely to be near one attractor or the other, but less likely to be in the middle. In this situation, the PDF has two “peaks”, and the mean and variance alone mean little.

9. Applications To Dynamical Systems (cont.)

10. Calculation of Polynomial Chaos Coefficients by Fast Unified Algorithms

11. Tools For Analysis of Dynamical Systems: Bifurcation Map of 4th order Runge Kutta ODE solver and a Basin Attractor Map for the Damped Forced Pendulum ODE via Interpolated Cell Mapping

12. In the case of Van Der Pol’s equation, pseudo-spectral routines performed poorly in calculating the 3rd order nonlinear term compared to the full-spectrum method. 8th degree PC was used solve in both variations. The plot at left shows that there is significant error in pseudo-spectral calculation with error in the 0th, 3rd, and 8th solution given. The plot at right shows that pseudo-spectral methods work poorly when all chaos coefficients are of similar magnitude. Even if magnitude decreases exponentially with chaos order, errors still appear.

14. Conclusions Polynomial Chaos is a promising tool for analysis of stochastic processes. More research into the stability of the Stochastic Collocation Method and related numerical aspects is necessary. A number of numerical and analytical challenges remain to be overcome before reliable application to complex dynamical systems.

15. References Xiu, D and Karniadakis, G. The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations, SIAM, 2002 Debusschere et. al, Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes, AMS, 2001 Heath, M.T. Scientific Computing, McGraw Hill, New York 2002 Abramowitz, M. and Stegun, I. A. (Eds.). "Orthogonal Polynomials." Ch. 22, Handbook of Mathematical Functions. New York: Dover, pp. 771-802, 1972.