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This thesis discusses various properties of a number of differential equations which we will term "integrable". There are many definitions of this word, but we will confine ourselves to two possible characterisations — either an equation can be transformed by a suitable change of variables to a linear equation, or there exists an infinite number of conserved quantities associated with the equation that commute with each other via some Hamiltonian structure. Both of these definitions rely heavily on the concept of the symmetry of a differential equation, and so Chapters 1 and 2 introduce and explain this idea, based on a geometrical theory of p.d.e.s, and describe the interaction of such methods with variational calculus and Hamiltonian systems. Chapter 3 discusses a somewhat ad hoc method for solving evolution equations involving a series ansatz that reproduces well-known solutions. The method seems to be related to symmetry methods, although the precise connection is unclear. The rest of the thesis is dedicated to the so-called Universal Field Equations and related models. In Chapter 4 we look at the simplest two-dimensional cases, the Bateman and Born-lnfeld equations. By looking at their generalised symmetries and Hamiltonian structures, we can prove that these equations satisfy both the definitions of integrability mentioned above. Chapter Five contains the general argument which demonstrates the linearisability of the Bateman Universal equation by calculation of its generalised symmetries. These symmetries are helpful in analysing and generalising the Lagrangian structure of Universal equations. An example of a linearisable analogue of the Born-lnfeld equation is also included. The chapter concludes with some speculation on Hamiltoian properties.

Thesis (Ph. D.)--University of Sydney, 2008. / Title from title screen (viewed December 12, 2008). Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy to the School of Mathematics and Statistics, Faculty of Science. Includes bibliographical references. Also available in print form.

M.Sc. (Mathematics) / Please refer to full text to view abstract

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Thesis (M.App.Sc.) - Carleton University, 2003. / Includes bibliographical references (p. 131-133). Also available in electronic format on the Internet.

Thesis (Master)--İzmir Institute of Technology, İzmir, 2005 / Keywords: Nonlinear differential equations, finite difference method, numeric simulation. Includes bibliographical references (leaves. 55-57).

Thesis (Ph. D.)--University of Washington, 2000. / Vita. Includes bibliographical references (leaves 142-145).

In this dissertation we focus on fractional-order dynamical systems and classify these problems as optimal control of system described by fractional derivative, fractional-order nonlinear differential equations, optimal control of systems described by variable-order differential equations and delay fractional optimal control problems. These problems are solved by using the spectral method and reducing the problem to a system of algebraic equations. In fact for the optimal control problems described by fractional and variable-order equations, the variables are approximated by chosen wavelets with unknown coefficients in the constraint equations, performance index and conditions. Thus, a fractional optimal control problem is converted to an optimization problem, which can be solved numerically. We have applied the new generalized wavelets to approximate the fractional-order nonlinear differential equations such as Riccati and Bagley-Torvik equations. Then, the solution of this kind of problem is found using the collocation method. For solving the fractional optimal control described by fractional delay system, a new set of hybrid functions have been constructed. Also, a general and exact formulation for the fractional-order integral operator of these functions has been achieved. Then we utilized it to solve delay fractional optimal control problems directly. The convergence of the present method is discussed. For all cases, some numerical examples are presented and compared with the existing results, which show the efficiency and accuracy of the present method.

In this thesis we study the behavior of a simple control system based on a delay differential equation with multiple loops of negative feedback. Numerical solutions of the delay differential equation with N delays d/dt x(t) = -x(t) + 1/N POT.N IND.i=1 / POT.n IND.i + x (t- IND.i) POT.n have been investigated as function of its parameters: n, i and i. A simple numerical method for determine the stability regions of the equilibrium points in the parameter space (i, n) is presented. The existence of a doubling period route to chaos in the equation, for N = 3, is characterized by the construction of bifurcation diagram with parameter n. A numerical method that uses the analysis of Poincaré sections of the reconstructed attractor to find aperiodic solutions in the parameter space of the equation is also presented. We apply this method for N = 2 and get evidences for the existence of chaotic solutions as result of a period doubling route to chaos (chaotic solutions for N = 2 in that equation had never been observed). Finally, we study the solutions of a piecewise constant equation that corresponds to the limit case n . / In this thesis we study the behavior of a simple control system based on a delay differential equation with multiple loops of negative feedback. Numerical solutions of the delay differential equation with N delays d/dt x(t) = -x(t) + 1/N POT.N IND.i=1 / POT.n IND.i + x (t- IND.i) POT.n have been investigated as function of its parameters: n, i and i. A simple numerical method for determine the stability regions of the equilibrium points in the parameter space (i, n) is presented. The existence of a doubling period route to chaos in the equation, for N = 3, is characterized by the construction of bifurcation diagram with parameter n. A numerical method that uses the analysis of Poincaré sections of the reconstructed attractor to find aperiodic solutions in the parameter space of the equation is also presented. We apply this method for N = 2 and get evidences for the existence of chaotic solutions as result of a period doubling route to chaos (chaotic solutions for N = 2 in that equation had never been observed). Finally, we study the solutions of a piecewise constant equation that corresponds to the limit case n .