On the oscillatory instability of multiple-parameter systems

The postcritical oscillatory behaviour of an autonomous discrete system under the influence of two independent parameters is studied. Three distinct situations are identified and explored via the intrinsic harmonic balancing technique introduced earlier. In each case, the asymptotic equations of the behaviour surface in parameter-amplitude space are derived explicitly. It is observed that there exists an interesting analogy between this surface and the equilibrium surface associated with static instabilities. Indeed, the phenomenon analyzed here is akin to fold catastrophe. The family of limit cycles associated with the behaviour surface are also obtained in general terms. The results can be used very easily to analyze specific problems, and this has been demonstrated on an illustrative example.