Let the system \[ \begin{aligned} dx/dt&=\rho x-\omega y+P(x,y,z;\mu) \\ dy/dt&=\omega x+\rho y+Q(x,y,z;\mu), \\ dz/dt&=\lambda z+R(x,y,z;\mu)\end{aligned}\tag{1} \] where \(\rho,\lambda >0\) and P, Q, R are analytic functions and \(\mu\) is a real parameter, possess a nontrivial solution \(\Gamma_ 0\) whose positive and negative limit sets consist solely of the origin. L. P. Shil’nikov [Sov. Math. Dokl. 6, 163-166 (1965); translation from Dokl. Akad. Nauk SSSR 160, 558-561 (1965; Zbl 0136.082) and Mat. Sb., n. Ser. 61(103), 443-466 (1963; Zbl 0121.075)] proved that every neighborhood of \(\Gamma_ 0\) possesses a denumerable collection of periodic solutions. The chaotic behaviour of the dynamical system of the type (1) is said to be chaotic in the sense of Shil’nikov. Namely, there is a homoclinic orbit at the origin and a “horseshoe” is embedded in a neighborhood of the homoclinic orbit. Hence there is a positively and negatively invariant Cantor set \(\Lambda\) containing: (i) infinitely many saddle-type (unstable) periodic orbits of arbitrary long periods; (ii) uncountably many bounded non-periodic orbits; (iii) a dense orbit. Moreover, the “horseshoe” persists under perturbation.
An oscillator in which chaos is conjectured to arise through a Shil’nikov mechanism has been studied by T. Matsumoto et. al. [T. Matsumoto, L. O. Chua, and K. Ayaki, IEEE Trans. Circuits Syst. CAS-35, No.7, 909-925 (1988; Zbl 0661.58019) and T. Matsumoto, L. O. Chua and M. Komuro, IEEE Trans. Circuits Syst. CAS-32, 797-818 (1985; Zbl 0578.94023)].
The authors of the paper under review carry out the investigation of chaotic behaviour arising through a Shil’nikov mechanism in modified Van der Pol oscillator. Their experimental results consist of a survey of the qualitative changes in the solution of the system describing Van der Pol oscillator, as a function of the control parameters. They find that the loci meet a codimension-2 point where lines of Hopf bifurcation intersect a locus of steady state bifurcations. The production of chaos then discussed in terms of finite-dimensional dynamical systems which are associated with such points. In particular, they identify a Takens- Bogdanov point and its associated line of homoclinic bifurcation.
Using the measured eigenvalue ratios, the Shil’nikov theory predicts a large, possibly infinite, number of coexisting attractors and a complex sequence of bifurcation, leading to chaos, as control parameters are varied. The authors’ experiments confirm this prediction by showing the existence of multiple coexisting states.