A classical scenario for tipping is that a dynamical system experiences a slow parameter drift across a fold tipping point, caused by a run-away positive feedback loop. We study what happens if one turns around after one has crossed the threshold. We derive a simple criterion that relates how far the parameter exceeds the tipping threshold maximally and how long the parameter stays above the threshold to avoid tipping in an inverse-square law to observable properties of the dynamical system near the fold. For the case when the dynamical system is subject to stochastic forcing we give an approximation to the probability of tipping if a parameter changing in time reverses near the tipping point. The derived approximations are valid if the parameter change in time is sufficiently slow. We demonstrate for a higher-dimensional system, a model for the Indian summer monsoon, how numerically observed escape from the equilibrium converge to our asymptotic expressions. The inverse-square law between peak of the parameter forcing and the time the parameter spends above a given threshold is also visible in the level curves of equal probability when the system is subject to random disturbances.

We consider networks of coupled scalar maps, with weighted connections which may include a time delay, and study the stability of equilibria with respect to the delays and connection structure. We prove that the largest eigenvalue of the graph Laplacian determines the effect of the connection topology on stability. The stability region in the parameter plane shrinks with increasing values of the largest eigenvalue, or of the time delay of the same parity. In particular, all bipartite graphs have an identical stability region, regardless of the delay or graph size, which is also the smallest stability region among those of all graphs. Furthermore, for certain parameter ranges, unstable (and possibly chaotic) maps can be stabilized via diffusive coupling with an odd time delay, provided that the network does not have a nontrivial and connected bipartite component. On the other hand, stabilization is not possible for even values of the delay or for bipartite networks.

We introduce a test for robustness of heteroclinic cycles that appear in neural microcircuits modeled as coupled dynamical cells. Robust heteroclinic cycles (RHCs) can appear as robust attractors in Lotka-Volterra-type winnerless competition (WLC) models as well as in more general coupled and/or symmetric systems. It has been previously suggested that RHCs may be relevant to a range of neural activities, from encoding and binding to spatio-temporal sequence generation.

For a dynamical system, it is known that the existence of a Lyapunov density implies almost global stability of an equilibrium. It is then natural to ask whether the existence of multiple Lyapunov densities for a nonlinear switched system implies almost global stability, in the same way as the existence of multiple Lyapunov functions implies global stability for nonlinear switched systems. In this work, the answer to this question is shown to be affirmative as long as switchings satisfy a dwell time constraint with an arbitrarily small dwell time. Specifically, as the main result, we show that a nonlinear switched system with a minimum dwell time is almost globally stable if there exist multiple Lyapunov densities which satisfy some compatibility conditions depending on the value of the minimum dwell time. This result can also be used to obtain a minimum dwell time estimate to ensure almost global stability of a nonlinear switched systems. In particular, the existence of a common Lyapunov density implies almost global stability for any arbitrary small minimum dwell time.

In this article, a sufficient condition on the minimum dwell time that guarantees the stability of switched linear systems is given. The proposed method interprets the stability of switched linear systems through the distance between the eigenvector sets of subsystem matrices. Thus, an explicit relation in view of stability is obtained between the family of the involved subsytems and the set of admissible switching signals.