Equation van der Pol

However, serious difficulties appeared later when efforts were made to attack more general problems not necessarily of the nearly-linear character. In terms of the van der Pol equation this occurs when the parameter is not small. Here the progress was far more difficult and the results less definite moreover there appeared two distinct theories, one of which was formulated by physicists along the lines of the theory of shocks in mechanics, and the other which was analytical and involved the use of the asymptotic expansions (Part IV of this chapter). The latter, however, turned out to be too complicated for practical purposes, and has not been extended sufficiently to be of general usefulness. 

This gives a simple graphical construction of integral curves. Suppose we wish to carry out this construction for the van der Pol equation for which f(x) = x2 — 1 F(x) = /3 — x. 

Nonanalytic Nonlinearities.—A somewhat different kind of nonlinearity has been recognized in recent years, as the result of observations on the behavior of control systems. It was observed long ago that control systems that appear to be reasonably linear, if considered from the point of view of their differential equations, often exhibit self-excited oscillations, a fact that is at variance with the classical theory asserting that in linear systems self-excited oscillations are impossible. Thus, for instance, in the van der Pol equation... 

The fluctuations of a self-exdted osdllator have been studied via a model based on the Van der Pol equation. Such an equation has been used to account for the ampUtude and phase fluctuations. A growth of the time coherence of phase above threshold is attributed to a decrease in an apparent diffusion coefficient for phase fluctuations. 

A less transparent example, but one that played a central role in the development of nonlinear dynamics, is given by the van der Pol equation... 

As you might guess, the system eventually settles into a self-sustained oscillation where the energy dissipated over one cycle balances the energy pumped in. This idea can be made rigorous, and with quite a bit of work, one can prove that the van der Pol equation has a unique, stable limit cycle for each fj >0. This result follows from a more general theorem discussed in Section 7.4. 

Show that the van der Pol equation has a unique, stable limit cycle. 

Hence condition (5) is satisfied for a = Vs. Thus the van der Pol equation has a unique, stable limit cycle, ... 

This analysis shows that the limit cycle has two widely separated time scales. the crawls require Ar O(/r) and the jumps require Ar Oiji ). Both time scales are apparent in the waveform of x(r) shown in Figure 7.5.2, obtained by numerical integration of the van der Pol equation for /r=10 and initial condition (Xo.yo) = (2.O). 

To illustrate the kinds of phenomena that can arise. Figure 7.6.1 shows a computer-generated solution of the van der Pol equation in the (x,x) phase plane, for e = 0.1 and an initial condition close to the origin, The trajectory is a slowly winding spiral it takes many cycles for the amplitude to grow substantially. Eventually... 

In Figure 7.6.4 we plot the exact solution of the van der Pol equation, obtained by numerical integration for =0.1 and initial conditions x(0) = l, x(0) = 0. For comparison, the slowly-varying amplitude r(T) predicted by (55) is also shown. The agreement is striking. Alternatively, we could have plotted the whole solution (56) instead of just its envelope then the two curves would be virtually indistinguishable, like those in Figure 7,6,3. ... 

In the non-linear systems (5.2), a second type of attractor — a closed curve (limit cycle) is also possible. For example, the system of van der Pol equations (representing oscillations of current in electrical circuits and oscillations of concentrations, or more precisely the differences between the concentrations and their stationary values, in chemical systems)... 

PitzHugh-Nagumo The FitzHugh-Nagumo equations are also called the Bonhoeffer-Van der Pol equations and have been used as a generic system that shows excitability and oscillatory activity. FitzHugh [1969] showed that much of the behavior of the Hodgkin-Huxley equations can be reproduced by a system of two differential equations ... 

One of the simplest nonlinear equation systems describing a circuit is the Van der Pol equation, which defines the oscillation of an unforced pendulum (Hairer and Wanner, 2010) ... 

On substitution, one will get the corresponding van der Pol equation for electric potential, assuming that the term K in Eq. (11.17) would be approximately constant, i.e. 

Quite complicated electric potential oscillations are observed in several phenomena involving (i) liquid-liquid, (ii) solid-liquid, (iii) solid-liquid and liquid-liquid and (iv) vapour-liquid interface (Chapter 11). Modelling has often been accomplished using Van der pol equation, a typical non-Unear equation. 

Here, the presented model is derived only for the linearized van der Pol equation, while the more realistic nonlinear case as well as calculated and experimentally corroborated data follow Refs. [37-50]. 

The bifurcation of a separatrix loop of a saddle-node was discovered by Andronov and Vitt [14] in their study of the transition phenomena from synchronization to beating modulations in radio-engineering. Specifically, they had studied the periodically forced van der Pol equation... 

Note that the effect of alternating zones of simple and complex behavior was discovered for the first time by van der Pol [154] in his experiments on the periodic forcing of a lamp generator (this effect occurs when one tunes a radio, and a characteristic noise is heard while moving from one station to another). The first theoretical explanation was given by Cartwright and Littlewood [36] for the van der Pol equation. 

The bifurcation of a limit cycle from the homoclinic loop to the saddle-node was first discovered by Andronov and Vitt in their study of the Van der Pol equation with a small periodic force at a 1 1 resonance ... 

Example 3.2 Consider again the van der Pol equation, (3.33). Alternative to the change of variable used in example 2.1, here we use the following change of variables ... 

The ability to solve nonlinear differential equations as readily as linear equations is one of the major advantages of the numerical solution of differential equations. For one such example, the Van der Pol equation is a classical nonlinear equation that has been extensively studied in the literature. It is defined in second order form and first order differential equation form as ... 

Figure 10.10. Example solution of nonlinear Van der Pol equation with a // = 20 parameter and with varying numbers of time steps.

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