Showing papers on "Infinite-period bifurcation published in 1980"

Secondary Bifurcation Near a Double Eigenvalue

Abstract: General conditions are formulated under which secondary bifurcation is rigorously established for a family of bifurcation problems depending continuously on a real auxiliary parameter. With more specific conditions, it is shown that, although the presence of secondary bifurcation renders the problem a priori degenerate, a full local bifurcation analysis is still possible.The results of this paper demonstrate the prime importance of symmetry (or more generally, invariance) to the mechanism by which secondary bifurcation points are created as the auxiliary parameter is varied.

Non-linear bifurcation analysis of non-gradient systems☆

Abstract: The post-divergence behaviour of non-gradient systems is studied through a perturbation approach, attention being restricted to equilibrium paths in the neighbourhood of a critical point. Simple and coincident critical points are treated separately. Various characteristic phenomena are explored in general terms by identifying certain distinct properties of the Jacobian matrix in a state-space formulation. It is demonstrated that the well-known asymmetric and symmetric points of bifurcation can also arise in non-gradient systems and the conditions giving rise to each of these phenomena are discussed. An illustrative example is presented.

Small Fluctuations in Systems with Multiple Limit Cycles

Abstract: We consider the effects of small random perturbations on deterministic systems of differential equations. The deterministic systems of interest have multiple limit cycles and may undergo a bifurcation (the Hopf bifurcation). We formulate a first exit problem for experiments beginning near stable and unstable limit cycles. The unstable limit cycle is surrounded by an annulus. Of interest is the probability of first exit from the annulus through a specified boundary, conditioned on initial position. The diffusion approximation is used, so that the conditional probability satisfies a backward diffusion equation. Approximate solutions of the backward equation are constructed by an asymptotic method. The behavior of the stochastic system in the vicinity of stable and unstable limit cycles is compared. When the deterministic system exhibits the Hopf bifurcation, the above analysis must be modified. Uniform solutions of the backward equation are constructed. Numerical examples are used to compare the theory with...

Transition from polar to duplicate patterns

Abstract: The existence of symmetric nonuniform solutions in nonlinear reaction-diffusion systems is examined. In the first part of the paper, we establish systematically the bifurcation diagram of small amplitude solutions in the vicinity of the two first bifurcation points. It is shown that: i) The system can adopt a stable symmetric solution (basic wave number 2) if the value of the bifurcation parameter is changed or if the initial polar structure (basic wave number 1) is sufficiently perturbed. ii) This behavior is independent of the particular reaction-diffusion model proposed and of the number of intermediate components (⩾2) involved.