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Steady-State/Hopf Interaction in the Van Der Pol Oscillator with Delayed Feedback

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Steady-State/Hopf Interaction in the Van Der Pol Oscillator with Delayed Feedback Steady-state/Hopf interaction in the van der Pol oscillator with delayed feedback Jason Bramburger Thesis submitted to the Faculty of Graduate and Postdoctoral Studies in partial fulfillment of the requirements for the degree of Master of Science in Mathematics 1 Department of Mathematics and Statistics Faculty of Science University of Ottawa c Jason Bramburger, Ottawa, Canada, 2013 1The M.Sc. program is a joint program with Carleton University, administered by the Ottawa- Carleton Institute of Mathematics and Statistics Abstract In this thesis we consider the traditional Van der Pol Oscillator with a forcing de- pendent on a delay in feedback. The delay is taken to be a nonlinear function of both position and velocity which gives rise to many different types of bifurcations. In particular, we study the Zero-Hopf bifurcation that takes place at certain parameter values using methods of centre manifold reduction of DDEs and normal form theory. We present numerical simulations that have been accurately predicted by the phase portraits in the Zero-Hopf bifurcation to confirm our numerical results and provide a physical understanding of the oscillator with the delay in feedback. ii Acknowledgements Thank you to my supervisors Dr. Victor Leblanc and Dr. Benoit Dionne for their guidance and support along the way and to the University of Ottawa for this oppor- tunity to further my education. iii Dedication This work is dedicated to my family for all of their support along the way. iv Contents List of Figures vii 1 Introduction 1 2 The Characteristic Equation 5 3 A Pure Imaginary Pair and a Simple Zero Eigenvalue 9 4 The Centre Manifold Reduction of DDEs 19 4.1 Necessary Terms in the Normal Form . 22 5 The Reduced System 26 6 Bifurcation Diagrams 44 7 Numerical Simulations 48 7.1 Simulation 1: The Nontrivial Fixed Point . 49 7.2 Simulation 2: The Stable Limit Cycle . 51 7.3 Simulation 3: The Invariant Torus . 53 8 Final Thoughts 60 A Calculation of the terms hijk 63 v CONTENTS vi B MATLAB Code 69 Bibliography 74 List of Figures 2.1 Diagram of the ("; a)-plane. The shaded area shows where complex conjugate roots of the characteristic equation occur. .6 3.1 The six topological types for the normal form given in (3:4). 11 3.2 The universal unfolding when b = +1, a > 0. On the curve µ2 = 2 2 µ1=a phase portraits are homeomorphic to those above the curve, but the lower saddle is degenerate if µ1 > 0 and the upper saddle is degenerate if µ1 < 0............................ 13 3.3 The universal unfolding when b = +1, a < 0. On the curve µ2 = 2 2 µ1=a phase portraits are topologically equivalent to those above the branches of the curve. The phase portrait at the origin is determined by the value of a; the left case being when a ≤ −1 and the right when a 2 (−1; 0). ............................ 15 3.4 The universal unfolding when b = −1, a < 0. On the curve µ2 = 2 2 µ1=a phase portraits are topologically equivalent to those below the branches of the curve. The phase portrait at the origin is determined by the value of a; the left case being when a 2 [−1; 0) and the right when a < −1. .............................. 16 vii LIST OF FIGURES viii 3.5 The universal unfolding when b = −1, a > 0. On the curve µ2 = 2 2 µ1=a phase portraits are topologically equivalent to those below the branches of the curve. 17 7.1 An unstable limit cycle bringing trajectories close then pushing them towards a nontrivial fixed point at approximately x = 0:3357. 50 7.2 (a) Multiple trajectories projected onto the centre manifold and con- verted to cylindrical coordinates and (b) the phase portrait predicted in Chapter 3. 51 7.3 A stable limit cycle in the system for (µ1; µ2) = (−0:02; 0:0495). 52 7.4 (a) Multiple trajectories projected onto the centre manifold and con- verted to cylindrical coordinates and (b) the phase portrait predicted in Chapter 3. The fixed point that trajectories converge to relates to a limit cycle in the original system. 53 7.5 A phase-locked periodic solution occurring on the stable invariant torus on the centre manifold when (µ1; µ2) = (−0:05; −0:2). 55 7.6 The one dimensional system showing a modulated oscillation relat- ing to a torus on the centre manifold. 55 7.7 (a) Projecting multiple trajectories of the system onto the centre manifold reveals a stable limit cycle in cylindrical coordinates and (b) The phase portrait predicted in Chapter 3. The limit cycle is given roughly by the dotted line. 56 7.8 A stable and unstable limit cycle within the invariant torus when (µ1; µ2) = (−0:02; −0:054). 58 Chapter 1 Introduction In 1920, while working as an engineer, Balthasar van der Pol published A theory of the amplitude of free and forced triode vibrations [15] in which he presented the oscillator that bears his name, given by x¨(t) + "(x2(t) − 1)x _(t) + x(t) = f(t); (1.1) where " > 0. Originally equation (1:1) was a model for an electrical circuit with a triode valve, and it has evolved into one of the most celebrated equations in the study of nonlinear dynamics. Van der Pol found that in the unforced system (when f ≡ 0) the trivial equilib- rium point is unstable and the system contains a stable oscillation for all values of ". He also explored the effect of a periodic forcing on his oscillator, which for example, can be given by 2πt x¨(t) + "(x2(t) − 1)x _(t) + x(t) = A cos : (1.2) T Upon adding this periodic forcing, van der Pol found that much more complicated phenomena could arise in the system. As the study of nonlinear differential equations has progressed, it has become known that the system exhibits chaotic behaviour when the nonlinearity becomes larger (i.e. " = 8:53). Because of the many different 1 1. Introduction 2 examples of nonlinear dynamics that occur in the system, the van der Pol oscillator has become a recurring example in modern mathematics. The van der Pol oscillator has become synonymous with modern systems that exhibit limit cycle oscillations in many far reaching branches of the physical sciences. Kaplan et al. provide a simple example of a biological application of the van der Pol oscillator to the dynamics of the heart [9], and the FitzHugh-Nagumo model is a modified form of the oscillator that has direct applications to neurones in the brain. On the larger scale, Cartwright et al. showed that the van der Pol oscillator can be used to model earthquake faults with viscous friction [3]. All of these applications and many more make the van der Pol oscillator one of the most widely celebrated differential equations in mathematics and one that is just as relevant today as the day that van der Pol himself presented the equation. Despite all of these applications of the van der Pol oscillator, it wasn't until around the start of the twenty-first century, almost eighty years after van der Pol introduced his equation, that mathematicians began to experiment with the effect of delays on the equation. Delay differential equations have proven to be useful in many physical problems ranging from the study of neural networks to the cutting of metal (see [13; 14]). For this reason, it is especially useful to study the effect of delays on one of the most famous nonlinear differential equations in all of mathematics to observe the power that a lag in communication can have over a dynamical system. Atay [2] studied the equation x¨(t) + "(x2(t) − 1)x _(t) + x(t) = "kx(t − τ); (1.3) where τ > 0 is a delay, and found that the stability of the limit cycle present in the system now depended on the amount of delay applied to the system. A similar study was presented by Oliveira [12], in which they added delays in the nonlinear terms of the unforced system, and found that periodic solutions still existed under certain conditions. Both of these studies confirmed that the effect of adding a delay 1. Introduction 3 to the equation can vastly affect the behaviour of the system and provide us with an outcome not easily predicted upon first inspection. It was Wei and Jiang [16; 17] who considered the forcing on the equation as a nonlinear function of the delay in position, x(t), given by x¨(t) + "(x2(t) − 1)x _(t) + x(t) = g(x(t − τ)): (1.4) They found that under certain conditions single, double and triple zero eigenvalues were possible as well as a purely imaginary pair with a zero eigenvalue. The cases of the single and double zero eigenvalues were presented in [17] and [8], respectively, with full details of the nature of their respective bifurcations. Later, Wu and Wang [19] undertook the bifurcation analysis of the purely imaginary pair and the zero eigenvalue to find that a Zero-Hopf bifurcation was occurring. They found that under certain conditions only two of the four possible bifurcation diagrams of the Zero-Hopf bifurcation were possible due to restrictions put in place by the initial assumptions on the nonlinear forcing. In this thesis we consider the van der Pol oscillator with general nonlinear delayed damping, x¨(t) + "(x2(t) − 1)x _(t) + x(t) = g(_x(t − τ); x(t − τ)); (1.5) 3 where g 2 C , g(0; 0) = 0, gx_ (0; 0) = a and gx(0; 0) = b, and work to understand its dynamics.
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