The Periodic Solution Of Van Der Pol Equation With Periodic Force

With the development of the science and technology, the relationship between Mathe-matics and other subjects becomes more and more close. For example, mathematical method is applied to electronic signal processing. As early as 1920s, a Dutch physicist named Balthasar Van der Pol investigated electrical circuits employing vacuum tubes and found that they have stable oscillations, then raised the "Van der Pol equation". From then on, we established the relationship between mathematics and electronics. In microwave circuit, Van der Pol oscillator can be used to model microwave negative resistance oscillator which is realized by diodes or bipolars [14,27,28]. Van der Pol oscillator model does not only make people understand the theory and design process of negative resistance oscillator more deeply, but also make people have a clear understanding of the phenomenon presented by the nonlinear microwave oscillator. In the study of mathematics and nonlinear dynamics, Van der Pol oscillator is often used to show some basic conceptions of nonlinear dynamical system, such as stability, limit cycle, Hopf bifurcation. It also plays an important role in the study of electricity.One type of Van der Pol oscillators is with periodic force, and this kind of oscillators play a vital role in the study of electronic signal control. In fact, the existence and uniqueness of a limit cycle for a periodically forced Van der Pol equation have been proved[18], and the relevant geometric approach has emerged. All of these works are related to asymptotic behavior of the solution as time goes to infinity. So far, the asymptotic solution of the Van der Pol equation with periodic external force has also been given in different ways[5,19,22], but almost all of these methods are relatively complicated, so how to get the asymptotic solution of equation simply and exactly is still one of the important and meaningful problems. In this paper, we use renormalization group method which is simple and effective to obtain the asymptotic periodic solution of equations in order to make further research in controlling electronic signal.We consider the circuit below, as the figure This circuit can be modeled by the following system of nonlinear differential equationswhere C is the capacitance,μis related to inductance, Vs is the input voltage, V is the output voltage,ψ(i) is the current-voltage characteristic of the circuit.Note thatψ(i) is piece-wise linear and is of the following form whereK1=Rf, K2=-(R2/R1)Rf and i0=-(R1/R2)V/Rf.The graph of the function is of single S-shaped.For the circuit we are interested, C andμare usually very small. The problem is thus a singularly perturbed problem with two small parameters. To ease the numerical computation and also the theoretical analysis we make a time variable transformation s=Ct. The differential Equations (1)then become whereε=μ/C, which we assumeεis relatively small. There are many practical applications of such nonlinear electric circuits in engineering. We are particularly interested in studying the relationship between the amplitude and frequency of a periodic input voltage Vs and the number of spikes of its output because different number of spikes corresponds to different signal. So we can achieve the purpose to control signal by controlling input voltage. If there is no input (Vs=0) andεis small, then system (2)has a stable limit cycle and its orbit in the phase plane will rotate around a limit cycle which is unique. For each rotation in the phase plane, the time series of the corresponding solution gives a spike or pulse asεis small. We call such solution a spike solution. More precise definition of the spike solution will be given in [4]. The formation of a spike solution can be intuitively described with a phase plane analysis.In this paper, we mainly consider the situation that the characteristic function is single S-shaped, then make use of renormalization group method which is one of the perturbation methods to get the uniformly valid asymptotic expansion solutions of the equations above and the time spent in emerging a spike.To make the notation more conventional we denote in the rest of this paper. And the system (2) becomesIn fact, (3)is a Van der pol osillator.This paper includes three chapters. The first chapter is an introduction. The second chapter briefly describes how to use renormalization group method which is established on the basis of the envelope theory to get the asymptotic solutions of differential equations. The third chapter is the main result in the condition that the characteristic function is single S-shaped, then make use of renormalization group method which is one of the perturbation methods to get the uniformly valid asymptotic expansion solutions of the equations above and the time spent in emerging a spike. Finally get the following theoremLemma 1 Assume that for periodic input f(t), system (3) has at least one-spike solu-tion, then the one-cycle time of the spike solution is given by where andp1, p2, P3, P4,q1, q2,q3,q4,r1,r3satisfy the following equations,respectively The number of spikes in one period of f(t) produced in the output voltage can be deter-mined roughly as n=[(t-)/(t0)]+1. Where r_ is the time f(t) spends in the region (i0,0) in one period of f(t). The formula for computing to has been given above.