arXiv subjects

Sebastian Wieczorek

Publications and source records attributed to Sebastian Wieczorek.

23 records · Page 2Linked to original sources

arXiv2016

Parameter shifts for nonautonomous systems in low dimension: Bifurcation- and Rate-induced tipping

We discuss the nonlinear phenomena of irreversible tipping for non-autonomous systems where time-varying inputs correspond to a smooth "parameter shift" from one asymptotic value to another. We express tipping in terms of pullback attraction and present some results on how nontrivial dynamics for non-autonomous systems can be deduced from analysis of the bifurcation diagram for an associated autonomous system where parameters are fixed. In particular, we show that there is a unique local pullback point attractor associated with each linearly stable equilibrium for the past limit. If there is a smooth stable branch of equilibria over the range of values of the parameter shift, the pullback attractor will remain close to (track) this branch for small enough rates, though larger rates may lead to rate-induced tipping. More generally, we show that one can track certain stable paths that go along several stable branches by pseudo-orbits of the system, for small enough rates. For these local pullback point attractors, we define notions of bifurcation-induced and irreversible rate-induced tipping of the non-autonomous system. In one-dimension, we give a number of sufficient conditions for the presence or absence of rate-induced tipping, and we discuss some applications of our results to give criteria for irreversible rate-induced tipping in a conceptual climate model example.

math.DS↗

arXiv2016

Three types of nonlinear resonances

We analyse different types of nonlinear resonances in a weakly damped Duffing oscillator using bifurcation theory techniques. In addition to (i) odd subharmonic resonances found on the primary branch of symmetric periodic solutions with the forcing frequency and (ii) even subharmonic resonances due to symmetry-broken periodic solutions that bifurcate off the primary branch and also oscillate at the forcing frequency, we uncover (iii) novel resonance type due to isolas of periodic solutions that are not connected to the primary branch. These occur between odd and even resonances, oscillate at a fraction of the forcing frequency, and give rise to a complicated resonance `curve' with disconnected elements and high degree of multistability. We use bifurcation continuation to compute resonance tongues in the plane of the forcing frequency vs. the forcing amplitude for different but fixed values of the damping rate. In this way, we demonstrate that identified here isolated resonances explain the intriguing structure of "patchy tongues" observed for week damping and link it to a seemingly unrelated phenomenon of "bifurcation superstructure" described for moderate damping.

math.DS↗

arXiv2014

Adapting to a Changing Environment: Non-obvious Thresholds in Multi-Scale Systems

Many natural and technological systems fail to adapt to changing external conditions and move to a different state if the conditions vary too fast. Such "non-adiabatic" processes are ubiquitous, but little understood. We identify these processes with a new nonlinear phenomenon---an intricate threshold where a forced system fails to adiabatically follow a changing stable state. In systems with multiple time-scales such thresholds are generic, but non-obvious, meaning they cannot be captured by traditional stability theory. Rather, the phenomenon can be analysed using concepts from modern singular perturbation theory: folded singularities and canard trajectories, including composite canards. Thus, non-obvious thresholds should explain the failure to adapt to a changing environment in a wide range of multi-scale systems including: tipping points in the climate system, regime shifts in ecosystems, excitability in nerve cells, adaptation failure in regulatory genes, and adiabatic switching in technology.

math.DS↗

arXiv2013

Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system

Tipping points associated with bifurcations (B-tipping) or induced by noise (N-tipping) are recognized mechanisms that may potentially lead to sudden climate change. We focus here a novel class of tipping points, where a sufficiently rapid change to an input or parameter of a system may cause the system to "tip" or move away from a branch of attractors. Such rate-dependent tipping, or R-tipping, need not be associated with either bifurcations or noise. We present an example of all three types of tipping in a simple global energy balance model of the climate system, illustrating the possibility of dangerous rates of change even in the absence of noise and of bifurcations in the underlying quasi-static system.

math.DS↗

arXiv2004

Bistable analytic-phase synchronization in strongly-competing chaotic lasing modes

We theoretically study analytic-phase synchronization in strongly-competing oscillator systems. Using the example of composite-cavity modes coupled via a class-B laser active medium, we discover that inherent chaotic phase synchronization can arise concurrently at two different chaotic attractors, leading to bistable phase-synchronized solutions. In our example, the underlying mechanism for bistability and inherent phase synchronization is population pulsation within the active medium.

nlin.CD↗