ME 406The van der Pol Oscillator‡1. IntroductionThe labels mathematician, engineer, and physicist have all been used in reference to Balthasar van der Pol.The van der Pol oscillator, which we study in this notebook, is a model developed by him to describe the behaviorof nonlinear vacuum tube circuits in the relatively early days of the development of electronics technology. Alittle more detail on his work, taken from the Exploratorium web site (http://www.exploratorium.edu), is givenbelow. A brief description of a circuit described by the van der Pol equation is given in Nonlinear Dynamics andChaos, Steven Strogatz, Addison-Wesley 1994, p. 228. Chapter 7 of Strogatz' book contains a very readablediscussion of the equation. Our study in this notebook will be based entirely on numerical solutions. The rigor-ous foundations for the analysis (e.g., the proof that the equation has a limit cycle solution which is a globalattractor) date back to the work of Lienard in 1928, with later more general analysis by Levinson and others.Perturbation techniques are also useful for the case of large parameter, but we will not consider them here.From Exploratorium web site:"Balthazar van der Pol was a Dutch electrical engineer who initiated modern experimental dynamics in thelaboratory during the 1920's and 1930's. Van der Pol investigated electrical circuits employing vacuum tubes andfound that they have stable oscillations, now called limit cycles. When these circuits are driven with a signalwhose frequency is near that of the limit cycle, the resulting periodic response shifts its frequency to that of thedriving signal.That is to say,the circuit becomes "entrained" to the driving signal. The waveform, or signal shape,however, can be quite complicated and contain a rich structure of harmonics and subharmonics. In the September1927 issue of the British journal Nature, he and his colleague van der Mark reported that an "irregular noise" washeard at certain driving frequencies between the natural entrainment frequencies. By reconstructing his electronictube circuit, we now know that they had discovered deterministic chaos. Their paper is probably one of the firstexperimental reports of chaos --- something that they failed to pursue in more detail. Van der Pol built a numberof electronic circuit models of the human heart to study the range of stability of heart dynamics. His investiga-tions with adding an external driving signal were analogous to the situation in which a real heart is driven by apacemaker. He was interested in finding out, using his entrainment work, how to stabilize a heart's irregularbeating or "arrhythmias"."Balthazar van der Pol (1889-1959)-------------------------------------------------------------Picture from Modern Differential Equations, Martha L. Abell andJames P. Braselton, Saunders, 1996.‡2. The van der Pol EquationThe van der Pol equation, in what is now considered to be standard form, is given by(1)x–+ m Hx2- 1L x°+ x = 0 .We see that it is an oscillator with a linear spring force and a nonlinear damping force. In all that follows, we takem > 0. The time in the equation has been scaled so that the frequency associated with the spring force alone isunity. The damping force varies in an interesting way. For »x» < 1, the damping is actually negative and henceproduces an amplification of the motion. For »x» > 1, there is true damping and the motion decays. These observa-tions suggest the possibility of an oscillation, in which the system starts at small x, is driven to large x by theamplification, and is then damped back to small x. We will explore this possibility by using DynPac to constructorbits. We define the equation for DynPac, after converting it to the following system:(2)x° = y, y° = -x - m(x2- 1)y .2 vanpol.nbsysidMathematica 5.2.0, DynPac 10.71, 10ê3ê2005intreset;plotreset;setstate@8x, y<D; setparm@8m<D; slopevec = 8y, -x - m Hx2- 1L y<;sysname = "van der Pol";For our initial explorations, we take the parameter m = 1.parmval = 81<;‡3. Equilibrium and StabilityThere is an equilibrium at the origin, and it is obvious from the slope vector that there are no other equilib -ria. We give this equilibrium a name, and then look at the eigenvalues of the derivative matrix eqmat at theequilibrium.eq = 80, 0<;eqmat = dermat ê. Thread@statevec Ø eqD880, 1<, 8-1, m<<Eigenvalues@eqmatD91ÅÅÅÅ2Im -è!!!!!!!!!!!!!!!!-4 + m2M,1ÅÅÅÅ2Im +è!!!!!!!!!!!!!!!!-4 + m2M=We see that if 0 < m < 2, the equilibrium is an unstable spiral. For m > 2, the equilibrium is an unstable node.Because there are no other equilibria, the only possible attractors for this system are the point at infinity, orperiodic solutions. We will let the computer tell us which. ‡4. The Limit CycleWe begin our numerical work with a phase portrait based on four selected initial conditions.initset = 880.5, 0<, 8-0.5, 0<, 83.0, 0<, 8-3.0, 0<<;We set the integration parameters.t0 = 0.0; h = 0.02; nsteps = 500;We set the plotting parameters.asprat = 1.0; plrange = 88-4, 4<, 8-4, 4<<; labshift = 15;vanpol.nb 3arrowflag = True; arrowvec = 81 ê 15<;graph1 = portrait@initset, t0, h, nsteps, 1, 2D;-4 -3 -2 -1 1 2 3 4x-4-3-2-11234yvan der Pol 8m<=8 1.00<This looks very much like a globally attracting limit cycle. Let's try to construct the pure limit cycle.sol2 = limcyc@82, 0<, t0, h, nstepsD;4 vanpol.nbgraph2 = phaser@sol2D;-4 -3 -2 -1 1 2 3 4x-4-3-2-11234yvan der Pol 8m<=8 1.00<[email protected] remarkable fact about this oscillator is that every initial condition in the phase plane ultimately leadsto this periodic motion with a period of 6.66. (Our calculations only suggest the truth of that statement. A proofrequires a rigorous mathematical analysis which is given in some of the references mentioned above.)‡5. Dependence of the Limit Cycle on the ParameterHow does the shape of this limit cycle change as the parameter m changes? We construct a sequence oflimit cycles with the parameter m varying from 0 to 3. We begin by looking at the two end-points of that m-rangeto get an idea of the period range and the range of function values in the graph. We first define a functioncycgraph[mval] which