Analysis of 2nd order differential equations : applications to chaos synchronisation and control

Johnson, Patrick (2008) Analysis of 2nd order differential equations : applications to chaos synchronisation and control. Doctoral thesis, Northumbria University.

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Abstract

In this thesis a number of open problems in the theory of ordinary differen-tial equations (ODEs) and dynamical systems are considered. The intention being to address current problems in the theory of systems control and synchronisation as well as enhance the understanding of the dynamics of those systems treated herein. More specifically, we address three central problems; the determination of exact analytical solutions of (non)linear (in)homogeneous ODEs of order 1 and 2, the de¬termination of upper/lower bounds on solutions of nonlinear ODEs and finally, the synchronisation of dynamical systems for the purposes of secure communication. With regard to the first of these problems we identify a new solvable class of Riccati equations and show that the solution may be written in closed-form. Fol¬lowing this we show how the Riccati equation solution leads us quite naturally to the identification of a new solvable class of 2nd order linear ODEs, as well as a yet more general class of Riccati equations. In addition, we demonstrate a new alter¬native method to Lagrange's variation of parameters for the solution of 2nd order linear inhomogeneous ODEs. The advantage of our approach being that a choice of solution methods is offered thereby allowing the solver to pick the simplest op¬tion. Furthermore, we solve, by means of variable transforms and identification of the first integral, an example of the Duffing-van der Pol oscillator and an associated ODE that connects the equations of Lienard and Riccati. These fundamental results are subsequently applied to the problem of solving the ODE describing a lengthen Abstract ii ing pendulum and the matter of bounded controller design for linear time-varying systems. In addressing the second of the above problems we generalise an existing GrOnwall-like integral inequality to yield several new such inequalities. Using one of the new inequalities we show that a certain class of nonlinear ODEs will always have bounded solutions and subsequently demonstrate how one can numerically evaluate the upper limits on the square of the solution of any given ODE in this class. Finally, we apply our results to an academic example and verify our conclu¬sions with numerical simulation. The third and final open problem we consider herein is concerned with the synchronisation of chaotic dynamical systems with the express intention of exploit¬ing that synchronisation for the purposes of secure transmission of information. The particular issue that we concern ourselves with is the matter of limiting the amount of distortion present in the message arriving at the receiver. Since the distortion encountered is primarily a due to the presence of noise and the message itself we meet our ends by employing an observer-based synchronisation technique incorpo¬rating a proportional-integral observer. We show how the PI observer used gives us the freedom to reduce message distortion without compromising on synchronisa¬tion quality and rate. We verify our results by applying the method to synchronise two parameter-matched Duffing oscillators operating in a chaotic regime. Simula tions clearly show the enhanced performance of the proposed method over the more traditional proportional observer-based approach under the same conditions. The structure of thesis is as follows: first of all we describe the motivation be¬hind object of study before going on to give a general introduction to the theory of ODEs and dynamical systems. This lead-in also includes a brief history of the the¬ory ODEs and dynamical systems, a general overview of the subject (as wholly as is possible without getting into the mathematical detail that is left to the appendices) and concludes with a statement of the scope of the thesis as well as the contribu¬tions to knowledge contained herein. We then go on to state and prove our main results and contributions to the solution of those problems detailed above starting with the solution of ODEs ...

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