拟周期可约鞍结分岔的数值计算：一种参数化方法途径

Interaction of two systems with saddle-node bifurcations on invariant circles. I. Foundations and the mutualistic case

The saddle-node bifurcation on an invariant circle (SNIC) is one of the codimension-one routes to creation or destruction of a periodic orbit in a continuous-time dynamical system. It governs the transition from resting behaviour to periodic spiking in many class I neurons, for example. Here, as a first step towards theory of networks of such units the effect of weak coupling between two systems with a SNIC is analysed. Two crucial parameters of the coupling are identified, which we call δ_1 and δ_2. Global bifurcation diagrams are obtained here for the "mutualistic" case δ_1 δ_2 > 0. Accordin…

Invariant Distributions and local theory of quasiperiodic cocycles in $\mathbb{T} ^{d} \times SU(2)$}

We study the linear cohomological equation in the smooth category over quasi-periodic cocycles in $\mathbb{T} ^{d} \times SU(2)$. We prove that, under a full measure condition on the rotation in $\mathbb{T} ^{d}$, for a generic cocycle in an open set of cocycles, the equation admits a solution for a dense set of functions on $\mathbb{T} ^{d} \times SU(2)$ of zero average with respect to the Haar measure. This property is known as Distributional Unique Ergodicity (DUE). We then show that given such a cocycle, for a generic function no such solution exists. We thus confirm in this context a conj…

Nonautonomous saddle-node bifurcations: random and deterministic forcing

We study the effect of external forcing on the saddle-node bifurcation pattern of interval maps. By replacing fixed points of unperturbed maps by invariant graphs, we obtain direct analogues to the classical result both for random forcing by measure-preserving dynamical systems and for deterministic forcing by homeomorphisms of compact metric spaces. Additional assumptions like ergodicity or minimality of the forcing process then yield further information about the dynamics. The main difference to the unforced situation is that at the critical bifurcation parameter, two alternatives exist. In …

The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations

We propose a general mechanism by which strange non-chaotic attractors (SNA) are created during the collision of invariant curves in quasiperiodically forced systems. This mechanism is first discussed on an heuristic level and by means of simulations. In the considered examples, a stable and an unstable invariant circle undergo a saddle-node bifurcation, but instead of a neutral invariant curve there exists a strange non-chaotic attractor-repeller pair at the bifurcation point. This process is accompanied by a very characteristic behaviour of the invariant curves prior to their collision, whic…

Merging and disconnecting resonance tongues in a pulsing excitable microlaser with delayed optical feedback

Excitability, encountered in numerous fields from biology to neurosciences and optics, is a general phenomenon characterized by an all-or-none response of a system to an external perturbation. When subject to delayed feedback, excitable systems can sustain multistable pulsing regimes, which are either regular or irregular time sequences of pulses reappearing every delay time. Here, we investigate an excitable microlaser subject to delayed optical feedback and study the emergence of complex pulsing dynamics, including periodic, quasiperiodic and irregular pulsing regimes. This work is motivated…

瞬态与网络相互作用通过捕获机制产生多稳态

Transients versus network interactions give rise to multistability through trapping mechanism

该研究揭示了耦合可兴奋单元网络中瞬态动力学与扩散耦合的竞争通过捕获机制产生多稳态，导致周期、准周期和混沌振荡共存，并阐明了吸引子经由鞍结或同宿分岔出现的机理。