The control of chaotic maps

**Authors:**Hoffman, Lance Douglas**Date:**2012-09-04**Subjects:**Control theory , Chaotic behavior in systems , Mappings (Mathematics)**Type:**Thesis**Identifier:**uj:3451 , http://hdl.handle.net/10210/6845**Description:**2003 , Some important ideas froni classical control theory are introduced with the intention of applying them to chaotic dynamical systems, in particular the coupled logistic equations. The structure of this dissertation is such that a strong foundation in control theory is first established before introducing the coupled logistic map or the methods of control and targetting in chaotic systems. In chapter 1 some aspects of classical control theory are reviewed. Continuous- and discrete-time dynamical systems are introduced and the existence and uniquendss criteria for the continuous case are explored via Lipschitz continuity. The matrix form of an inhomogeneous linear differential equation is presented and several properties of the associated transition matrix are discussed. Several linear algebraic ideas, most notably the Cayley-Hamilton theorem, are employed to explore the important concepts of controllability and observability in linear systems. The stabilisability problem is thoroughly investigated. Finally, the neighbourhood properties of continuous nonlinear dynamical systems with reference to controllability, stability and noise are established. Chapter 2 places emphasis on canonical forms, pole assignments and state observers. The decomposition of a general system into distinct components is facilitated by the general structure theorem, which is proved. The pole placement problem is described and the correspondence between the stabilisability of a system and the placement of poles is noted by the use'of a socalled feedback matrix. Lastly, the notion of a state observer, with reference to some dynamic feedback law, is introduced. The dynamics of the coupled logistic equations are studied in chapter 3. The fixed points of the map are calculated and the subsequent dynamical consequences explored. Using methods introduced in earlier chapters, the stability of the map is investigated. Using the so-called variational equations, the Lyapunov exponents are computed and used to classify, the motion of the system for the parameter values r and a. This chapter concludes with a discussion of the basins of attraction and critical curves associated with the coupled logistic equations. It is in chapter 4 that the models for controlling chaos are instantiated. The famous Ott-Grebogi- Yorke (OGY) method for controlling chaos is explained and related to the pole placement problem, discussed previously. The theory is extended to study the control of periodic orbits with periods greater than one.**Full Text:**

**Authors:**Hoffman, Lance Douglas**Date:**2012-09-04**Subjects:**Control theory , Chaotic behavior in systems , Mappings (Mathematics)**Type:**Thesis**Identifier:**uj:3451 , http://hdl.handle.net/10210/6845**Description:**2003 , Some important ideas froni classical control theory are introduced with the intention of applying them to chaotic dynamical systems, in particular the coupled logistic equations. The structure of this dissertation is such that a strong foundation in control theory is first established before introducing the coupled logistic map or the methods of control and targetting in chaotic systems. In chapter 1 some aspects of classical control theory are reviewed. Continuous- and discrete-time dynamical systems are introduced and the existence and uniquendss criteria for the continuous case are explored via Lipschitz continuity. The matrix form of an inhomogeneous linear differential equation is presented and several properties of the associated transition matrix are discussed. Several linear algebraic ideas, most notably the Cayley-Hamilton theorem, are employed to explore the important concepts of controllability and observability in linear systems. The stabilisability problem is thoroughly investigated. Finally, the neighbourhood properties of continuous nonlinear dynamical systems with reference to controllability, stability and noise are established. Chapter 2 places emphasis on canonical forms, pole assignments and state observers. The decomposition of a general system into distinct components is facilitated by the general structure theorem, which is proved. The pole placement problem is described and the correspondence between the stabilisability of a system and the placement of poles is noted by the use'of a socalled feedback matrix. Lastly, the notion of a state observer, with reference to some dynamic feedback law, is introduced. The dynamics of the coupled logistic equations are studied in chapter 3. The fixed points of the map are calculated and the subsequent dynamical consequences explored. Using methods introduced in earlier chapters, the stability of the map is investigated. Using the so-called variational equations, the Lyapunov exponents are computed and used to classify, the motion of the system for the parameter values r and a. This chapter concludes with a discussion of the basins of attraction and critical curves associated with the coupled logistic equations. It is in chapter 4 that the models for controlling chaos are instantiated. The famous Ott-Grebogi- Yorke (OGY) method for controlling chaos is explained and related to the pole placement problem, discussed previously. The theory is extended to study the control of periodic orbits with periods greater than one.**Full Text:**

Gravitational capture

**Authors:**Anderson, Keegan Doig**Date:**2012-11-02**Subjects:**Gravitation , Space trajectories , Dynamics , Three-body problem , Two-body problem**Type:**Thesis**Identifier:**http://ujcontent.uj.ac.za8080/10210/365597 , uj:7315 , http://hdl.handle.net/10210/8053**Description:**M.Sc. , Important ideas from dynamical systems theory and the restricted three-body problem are introduced. The intention is the application of dynamical systems theory techniques to the restricted three-body problem to better understand the phenomenon of gravitational capture. Chapter 1 gives a much deeper review of the purpose of this dissertation. Chapter 2 introduces and reviews important concepts from dynamical systems. Chapter 3 reviews the restricted three-body problem and all important aspects of the problem. In chapter 4 we define and study the phenomenon of gravitational capture. We take a novel approach by applying a symplectic method, namely the implicit midpoint method, to model trajectories in the restricted three-body problem. As far as we know, this is the first time such a method has actually been applied, with other authors preferring to apply explicit methods in trajectory modelling. In the closing of this chapter we review our whole discourse and suggest topics for future research. The disseration is concluded with two appendix chapters. In the first chapter we list all the computer code we have written for this dissertation. The second appendix chapter reviews the n-body problem and we show a full solution of the two-body problem.**Full Text:**

**Authors:**Anderson, Keegan Doig**Date:**2012-11-02**Subjects:**Gravitation , Space trajectories , Dynamics , Three-body problem , Two-body problem**Type:**Thesis**Identifier:**http://ujcontent.uj.ac.za8080/10210/365597 , uj:7315 , http://hdl.handle.net/10210/8053**Description:**M.Sc. , Important ideas from dynamical systems theory and the restricted three-body problem are introduced. The intention is the application of dynamical systems theory techniques to the restricted three-body problem to better understand the phenomenon of gravitational capture. Chapter 1 gives a much deeper review of the purpose of this dissertation. Chapter 2 introduces and reviews important concepts from dynamical systems. Chapter 3 reviews the restricted three-body problem and all important aspects of the problem. In chapter 4 we define and study the phenomenon of gravitational capture. We take a novel approach by applying a symplectic method, namely the implicit midpoint method, to model trajectories in the restricted three-body problem. As far as we know, this is the first time such a method has actually been applied, with other authors preferring to apply explicit methods in trajectory modelling. In the closing of this chapter we review our whole discourse and suggest topics for future research. The disseration is concluded with two appendix chapters. In the first chapter we list all the computer code we have written for this dissertation. The second appendix chapter reviews the n-body problem and we show a full solution of the two-body problem.**Full Text:**

A numerical method based on Runge-Kutta and Gauss-Legendre integration for solving initial value problems in ordinary differential equations

- Prentice, Justin Steven Calder

**Authors:**Prentice, Justin Steven Calder**Date:**2012-09-11**Subjects:**Differential equations - Numerical methods**Type:**Thesis**Identifier:**uj:9930 , http://hdl.handle.net/10210/7329**Description:**M.Sc. , A class of numerical methods for solving nonstiff initial value problems in ordinary differential equations has been developed. These methods, designated RKrGLn, are based on a Runge-Kutta method of order r (RKr), and Gauss-Legendre integration over n+ 1 nodes. The interval of integration for the initial value problem is subdivided into an integer number of subintervals. On each of these n + 1 nodes are defined in accordance with the zeros of the Legendre polynomial of degree n. The Runge-Kutta method is used to find an approximate solution at each of these nodes; Gauss-Legendre integration is used to find the solution at the endpoint of the subinterval. The process then carries over to the next subinterval. We find that for a suitable choice of n, the order of the local error of the Runge- Kutta method (r + 1) is preserved in the global error of RKrGLn. However, a poor choice of n can actually limit the order of RKrGLn, irrespective of the choice of r. What is more, the inclusion of Gauss-Legendre integration slightly reduces the number of arithmetical operations required to find a solution, in comparison with RKr at the same number of nodes. These two factors combine to ensure that RKrGLn is considerably more efficient than RKr, particularly when very accurate solutions are sought. Attempts to control the error in RKrGLn have been made. The local error has been successfully controlled using a variable stepsize strategy, similar to that generally used in RK methods. The difference lies in that it is the size of each subinterval that is controlled in RKrGLn, rather than each individual stepsize. Nevertheless, local error has been successfully controlled for relative tolerances ranging from 10 -4 to 10-10 . We have also developed algorithms for estimating and controlling the global error. These algorithms require that a complete solution be obtained for a specified distribution of nodes, after which the global error is estimated and then, if necessary, a new node distribution is determined and another solution obtained. The algorithms are based on Richardson extrapolation and the use of low-order and high-order pairs. The algorithms have successfully achieved desired relative global errors as small as 10-1° . We have briefly studied how RKrGLn may be used to solve stiff systems. We have determined the intervals of stability for several RKrGLn methods on the real line, and used this to develop an algorithm to solve a stiff problem. The algorithm is based on the idea of stepsize/subinterval adjustment, and has been used to successfully solve the van der Pol system. Lagrange interpolation on each subinterval has been implemented to obtain a piecewise continuous polynomial approximation to the numerical solution, with same order error, which can be used to find the solution at arbitrary nodes.**Full Text:**

**Authors:**Prentice, Justin Steven Calder**Date:**2012-09-11**Subjects:**Differential equations - Numerical methods**Type:**Thesis**Identifier:**uj:9930 , http://hdl.handle.net/10210/7329**Description:**M.Sc. , A class of numerical methods for solving nonstiff initial value problems in ordinary differential equations has been developed. These methods, designated RKrGLn, are based on a Runge-Kutta method of order r (RKr), and Gauss-Legendre integration over n+ 1 nodes. The interval of integration for the initial value problem is subdivided into an integer number of subintervals. On each of these n + 1 nodes are defined in accordance with the zeros of the Legendre polynomial of degree n. The Runge-Kutta method is used to find an approximate solution at each of these nodes; Gauss-Legendre integration is used to find the solution at the endpoint of the subinterval. The process then carries over to the next subinterval. We find that for a suitable choice of n, the order of the local error of the Runge- Kutta method (r + 1) is preserved in the global error of RKrGLn. However, a poor choice of n can actually limit the order of RKrGLn, irrespective of the choice of r. What is more, the inclusion of Gauss-Legendre integration slightly reduces the number of arithmetical operations required to find a solution, in comparison with RKr at the same number of nodes. These two factors combine to ensure that RKrGLn is considerably more efficient than RKr, particularly when very accurate solutions are sought. Attempts to control the error in RKrGLn have been made. The local error has been successfully controlled using a variable stepsize strategy, similar to that generally used in RK methods. The difference lies in that it is the size of each subinterval that is controlled in RKrGLn, rather than each individual stepsize. Nevertheless, local error has been successfully controlled for relative tolerances ranging from 10 -4 to 10-10 . We have also developed algorithms for estimating and controlling the global error. These algorithms require that a complete solution be obtained for a specified distribution of nodes, after which the global error is estimated and then, if necessary, a new node distribution is determined and another solution obtained. The algorithms are based on Richardson extrapolation and the use of low-order and high-order pairs. The algorithms have successfully achieved desired relative global errors as small as 10-1° . We have briefly studied how RKrGLn may be used to solve stiff systems. We have determined the intervals of stability for several RKrGLn methods on the real line, and used this to develop an algorithm to solve a stiff problem. The algorithm is based on the idea of stepsize/subinterval adjustment, and has been used to successfully solve the van der Pol system. Lagrange interpolation on each subinterval has been implemented to obtain a piecewise continuous polynomial approximation to the numerical solution, with same order error, which can be used to find the solution at arbitrary nodes.**Full Text:**

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