Nonlinear stochastic dynamics and chaos by numerical path integration Eirik Mo January 5, 2008 Acknowledgements The main gratitude goes to Professor Arvid Naess, who has been the advisor for this work, and given many valuable comments. The problem description was written by Naess, the development of path integration for stochastic differential equations is largely devoted to him, and he also applied for and received funding for the project. For four years, I have worked together with Dr. Hans Christian Karlsen, who also was a PhD student advised by Naess. Karlsen's work on running path integration for some specific problems was of great importance for the development of more stable and efficient code. His validation of accuracy and general comments have been very helpful for this work. From August 2004 until February 2005, I visited Professor Mircea Grigoriu at the Department of Structural Engineering at Cornell University, beautifully situated in Ithaca by the Finger Lakes, upstate New York. This was a wonderful stay, where I were warmly welcomed by PhD students at the department, and a lot of people in the area. I would especially like to mention Ms. (Kathleen) Misti Wilcox, who rented out rooms in her house, cooked wonderful meals, proudly presented Ithaca and the surrounding area, and motivated me for working. This visit was a big inspiration, and a turning point in the research. The Research Council of Norway funded the project and the visit to Cornell University, of which I'm very grateful. Motivating words from people around me have been important from time to time. I cannot mention everyone by name, but they basically consist of family, my colleagues at the Department of Mathematical Sciences, my best friends from the master programme who meet at the \cantina", the \ISFiT 2001 contact group", the interest organization for doctoral candidates at NTNU (DION), my housing cooperative, friends from charity organizations, and all my friends in the US. I would particularly like to thank my mother Linn Mo for reading through the thesis and correcting some basic language errors. i ii iii Preface The main purpose of this thesis is to study Stochastic Differential Equations (SDEs) using Path Integration (PI). Nonlinear dynamical systems is a main focus of this work, both because such systems present important challenges for the PI method, and because these systems are generally hard to analyze analytically and by means of other numerical meth- ods. More specifically, three kinds of systems will be discussed; simpler test cases where analytical solutions exist, systems that exhibit chaotic response, and systems with instan- taneous changes at fixed values for the variable. The last group consists of signum-kind nonlinearities, impact systems with completely elastic impact and contacting impacts, and absorbing boundaries in reliability studies. For some systems, especially the chaotic sys- tems, the random noise is basically a way to model small perturbations of the system. The role of this noise is to make theory and numerical methods from stochastic processes applicable, to model imperfections or noise in a physical system, or to account for the un- certainty in parameter estimates in such models. In other systems, the noise could be the main driving force, without which the system would just converge to one stability point. A lot of this work is based on research from three earlier PhD candidates, [20, 34, 50]. Some of the techniques developed there are continued, some observations are investigated more thoroughly, and some of the problems and challenges found in this process will be discussed and explained. This also means that some important issues like convergence, stability, choice of numerical methods etc. have already been answered, so only the con- clusions are referred here. In the beginning of this work, the main question was if PI could be used as a tool to study chaotic systems, and if so, in what sense. One of the main characteristics of chaotic systems are \strange attractors"; limit sets with a complicated structure and fractal dimen- sion. When adding noise, these limit sets are in some ways smeared out in the probability density over the state space, but this density could still have a very rugged structure. This is one of the challenges for PI, which requires a good approximative representation of the probability density by some interpolating surface. Different ways to perform a stable in- terpolation is discussed, and it is shown that the choice of interpolation could be crucial for the accuracy and convergence of the method, even for much simpler problems. The interpolation problem is related to the problem of discretizing the state space variables. Interpolation error seems to be the main reason for unstable results in earlier work, so this is the first challenge discussed in this thesis, and requires close attention from those who plan to follow up this method further. All interpolation methods used here have been written by the author, and are modified to attain the required stability and accuracy for the problems discussed. So this work is new. The second challenge is the nonlinearity of the system, which is extremely important to handle properly. Time stepping methods like the Euler method, Runge Kutta schemes, and Taylor will be discussed together with the problem of choosing the optimal time step length. The path integration itself also requires that the nonlinear effects are properly taken care of. Except for the Taylor method, the programming code for the time stepping method is written by the author, though using standard methods. A reason that library functions iv are not used, is that the time stepping method is highly integrated in the rest of the code to allow for optimization when many points are propagated forwards simultaneously. A third challenge is dimensionality. Solving a one-dimensional autonomous SDE is fairly trivial and can be done with high accuracy in short time. A differential equation with chaotic response will either have three or more dimensions, or contain terms that explicitely depend upon time, e.g. periodic forcing. It is also of interest to study sys- tems of coupled oscillators, filtered noise processes etc., which gives systems with a higher number of dimensions. Not only will computational time and memory requirements grow exponentially with the number of dimensions; even writing the computer code for such systems soon becomes an extremely challenging task. How do you write efficient computer code for a given problem with any number of dimensions and any choice of approximation method - without typing errors? One solution is to let the computer code be generated by yet another program, and this is a new development. This required describing the method of Path Integration and all its numerical structures in a generalized way. Some of the mathematical observations in this generalization and methods used to create efficient code is presented in this book. A milestone for the PI method, was the implementation and running of a stable system with six dimensions (6D), which was finally accomplished in April 2007. Six dimensions may not seem to be a lot for researchers from other fields, but to the authors' knowledge a full 6D response PDF of an SDE has not been computed by any numerical method earlier. It has not been a main focus to do detailed studies of a large number of differential equations with chaotic and non-chaotic response under the influence of various noise pro- cesses. This is a vast field with endless possibilities, and there are many articles discussing various topics related to this - often for some very specific differential equations. Partly because of the challenges described above, there is very little work performed earlier that allows for detailed comparizations to the results of the PI method. Some theory on chaos, chaotic and non-chaotic strange attractors is included. In addi- tion, some of the varieties of possible behaviour when such systems are affected by noise is described, and illustrated by the PI method. PI could also give more insight to the behaviour of a system through other statistics than the PDF. Since chaos was a main in- terest, a great deal of work has been done trying to obtain the maximal Lyapunov exponent (MLE) of a system. Also here, the main question is if the PI method can be used for this task. The MLE, and the Lyapunov spectrum, is not trivial to compute with the methods already proposed by other authors - for general systems. There are many different kinds of questions that could be asked about adding noise to a differential equation. In some cases, the problem is stability of the attractor, and the noise is just there to create a small perturbation kicking the system off from the deterministic path. For other applications, the qualities of the noise process could be of vital importance, and different noise levels or noise processes could give very different behaviour of the system. The latter case requires knowing that the numerical method can handle the stochastic terms with sufficient accuracy. In cooperation with professor Mircea Grigoriu at Cornell University, this question developed into some studies of the compound Poisson process, and the possibilities of implementing compound Poisson noise in PI. A master student at v NTNU, Tore Selland Kleppe, studied the implementation of L´evy processes in PI, partly in correspondance with the author [26]. The results from these attempts to use the PI method in other fields will however not be discussed in this work, due to space limitations. Many methods in the field of stochastic dynamical systems are limited to processes with Gaussian increments. In the implementation of PI, a main focus is to avoid this restraint. Comparing PI to other numerical algorithms is also beyond the scope of this study, but a brief description of some other methods can be found in sections 2.2 and 4.5.