# Van der Pol method in a quasilinear equation with multiple fixed points within a cycle.

My question is about details of application of the van der Pol - Andronov method to analysis of quasilinear ordinary differential equations. Before formulating the question, let me first give necessary explanations to avoid any misunderstanding. Quasilinear ordinary differential equation is the one having the form :

$$z''(t) + z(t) = f(z(t),z'(t)) \qquad \qquad (1)$$

where $t$ is the time, $z=z(t)$ is the unknown variable, and $f(z,z')$ is a nonlinear function, which is small for whatever reason. In this case in order to analyze the phase portrait of such an equation one can apply the so - called, van der Pol - Andronov method. In short, this method averages the equation (1) over a phase variable, which enables one to establish a relation between details of the phase portrait of the equation (1) (such as a node, or a focus in the coordinates origin, and limit cycles, if any) to the number and positions of fixed points of a simpler equation:

$$r'=F(r) \qquad \qquad (2)$$

where $r=r(t)$ is a new "radial" variable, and the nonlinear function $F=F(r)$ is obtained from the function $f(z, z')$ by the mentioned averaging. Equation (2) may only have one type of attractors (if any), that are the fixed points. According to the way of mapping of (2) onto (1) the fixed point at $r=0$ of Eq. (2) is interpreted as a focus or a node of the initial equation (1), while its nonzero fixed point - as a limit cycle of equation (1). The method is known over a century, described in textbooks, as for example, Andronov,A.A.,Vitt,A.A.& Khaikin,S.E.Theory of Oscillators, Dover Publications, 1987 and it is well grounded.

It is quite clear, however, if (1) has only a single focus or a single node located within one or more cycles, or only a focus or node with no cycles at all, or no attractors at all.

Assume now that a quasilinear equation (1) has a solution with several fixed points in the vicinity of the coordinate origin, which are enclosed within one or few cycles. It seems clear that Eq. (2) is unable to resolve several fixed points. Am I wrong with this statement?

My question is as follows. Is the van der Pol-Andronov approach still applicable to such a situation? May it be that it is applicable partially, say, only for detection of the cycles, though it cannot "resolve" the fixed points? Could you kindly point out any textbook or any other literature treating any example of such a situation?

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