Experimental Evidences of Shil'nikov Chaos and Mixed-mode Oscillation in Chua Circuit

1. Introduction

Homoclinic orbit is asymptotic to a saddle limit set both forward and backward in time (Wiggins, 1990; Kuznetsov, 1995). In the vicinity of a homoclinic orbit, if a control parameter is tuned, a countable infinity of periodic orbits are observed, which are at the origin of chaos in nonlinear dynamical system. The homoclinic orbit is structurally unstable and not possible to observe either in numerical or physical experiments. However, homoclinic chaos of the Shilnikov type has been observed, in the vicinity of the homoclinic orbit, by tuning a suitable control parameter in BZ reaction (Petrov et al, 1992), liquid crystal flow (Peacock & Mullin, 2001), CO₂ laser (Pisarchik et al,2001; Allaria et al, 2001), optothermal bistable device (Herrero et al, 1996) and electronic circuit (Healy et al, 1996). The trajectory of homoclinic chaos is globally stable yet instabilities are bounded to a local domain close to a saddle, which may be a saddle focus or a saddle cycle in 3D system. The local instability has its manifestation in large fluctuations in the return time of spiking oscillation. It is to be noted that homoclinic bifurcation is considered, in recent times (Belykh et al, 2000; Izhikevich, 2000), as one of the important mechanisms of emergence of the spiking and bursting behaviors in neurons with inherent fast and slow dynamics.

The Shil'nikov chaos (Wiggins, 1990; Kuznetsov, 1995) deals with a saddle focus with real and complex conjugate eigenvalues, (γ, σ±jω) in 3D systems. The trajectory of the homoclinic chaos escapes spirally from the saddle focus in 2D eigensapce and re-injects into it along the stable eigendirection for systems with γ<0, σ>0 and |γ/σ|>1. A reverse direction of the trajectory of the homolcinic chaos is seen when γ>0, σ<0 and |γ/σ|>1. However, it is true, in general, that in period-parameter space, the period of a limit cycle (period-n: n⁰, n=1, 2, 3…) increases asymptotically with a control parameter as it approaches the homoclinic orbit or the bifurcation point and in close vicinity of this bifurcation point, instabilities appear yet bounded to a saddle focus which is defined as Shil’nikov chaos. Further studies (Glendinning & Sparrow, 1984) on Shil'nikov chaos show that a Shil'nikov wiggle (Wiggins, 1990) may appear in period-parameter space if 1/2<|γ/σ|<1 for γ<0 and σ>0. Under this condition, when a control parameter of a system is tuned from both sides of the homoclinic point, the period of a limit cycle increases in a wiggle with alternate sequences of stable and unstable orbits via saddle-node (SN) and period-doubling (PD) bifurcations respectively. The Shil'nikov wiggle is beyond the scope of this report. We restrict our discussion here on simpler condition for Shil'nikov chaos,|γ/σ|>1, when we find sequences of mixed mode oscillation (MMO) and homoclinic bifurcation.

In many nonlinear dynamical systems such as Chua circuit, the dynamics usually changes with a parameter from stable equilibrium to limit cycle by super-Hopf bifurcation and to chaos via (PD). With further changes in parameter, the system shows period-adding bifurcation when a sequence of periodic windows appear intermediate to chaotic windows in parameter space. The periodic windows are created via SN bifurcation of the chaotic behavior while the periodic states again move to chaotic states via PD. Subsequently, the dynamics follows a reverse PD before moving to period-1 and then to unstable limit cycle via subcritical Hopf bifurcation.