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We consider dissipative periodically forced systems and investigate cases in which having information as to how the system behaves for constant dissipation may be used when dissipation varies in time before settling at a constant final…
Non-autonomous dynamical systems help us to understand the implications of real systems which are in contact with their environment as it actually occurs in nature. Here, we focus on systems where a parameter changes with time at small but…
We consider unstable attractors; Milnor attractors $A$ such that, for some neighbourhood $U$ of $A$, almost all initial conditions leave $U$. Previous research strongly suggests that unstable attractors exist and even occur robustly (i.e.…
Identification of attractors, that is, stable states and sustained oscillations, is an important step in the analysis of Boolean models and exploration of potential variants. We describe an approach to the search for asynchronous cyclic…
External and internal factors may cause a system's parameter to vary with time before it stabilizes. This drift induces a regime shift when the parameter crosses a bifurcation. Here, we study the case of an infinite dimensional system: a…
We present a dynamical system that naturally exhibits two unstable attractors that are completely enclosed by each others basin volume. This counter-intuitive phenomenon occurs in networks of pulse-coupled oscillators with delayed…
Attractors in parity games are a technical device for solving "alternating" reachability of given node sets. A well known solver of parity games - Zielonka's algorithm - uses such attractor computations recursively. We here propose new…
In this paper, we demonstrate, first in literature known to us, that potential functions can be constructed in continuous dissipative chaotic systems and can be used to reveal their dynamical properties. To attain this aim, a Lorenz-like…
We propose a theoretical framework for an explanation of the numerically discovered phenomenon of the attractor-repeller merger. We identify regimes which are observed in dynamical systems with attractors as defined in a work by Ruelle and…
This work contains the results from a comprehensive study of a new class of attractors. The attractors in this class are characterized by strong local instability, but they are not uniformly hyperbolic. Rigorous results on their dynamical,…
The chaotic properties of Newton-Leipnik system are discussed from the view point of strange attractors. Previously, two strange attractors of this system were illustrated which occured from two different initial conditions under the same…
How long does a trajectory take to reach a stable equilibrium point in the basin of attraction of a dynamical system? This is a question of quite general interest, and has stimulated a lot of activities in dynamical and stochastic systems…
We present a fully automated method that identifies attractors and their basins of attraction without approximations of the dynamics. The method works by defining a finite state machine on top of the system flow. The input to the method is…
This paper investigates two issues on identification of switched linear systems: persistence of excitation and numerical algorithms. The main contribution is a much weaker condition on the regressor to be persistently exciting that…
We have investigated noise-induced transition of atoms between double or triple phase-space attractors that are produced in the parametrically driven magneto-optical trap. The transition rates between two or three dynamic attractors,…
A one-parameter family of time-reversible systems on $\mathbb{T}^3$ is considered. It is shown that the dynamics is not conservative, namely the attractor and repeller intersect but not coincide. We explain this as the manifestation of the…
Slow parameter drift is common in many systems (e.g., the amount of greenhouse gases in the terrestrial atmosphere is increasing). In such situations, the attractor on which the system trajectory lies can be destroyed, and the trajectory…
The purpose of this note is threefold. First we state a few conjectures that allow us to rigorously derive a theory which is asymptotic in N (the number of agents) that describes transients in large arrays of (identical) linear damped…
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…
In a dissipative system, there exists the (global) attractor which has finite fractal dimensions. The flow on the attractor can be parametrized by a finite number of parameters (Temmam 1987). Using machine learning we demonstrate how to…