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In order to investigate the evolutionary process of many deterministic Dynamical systems with unfixed parameter, a set of dynamical models with parameter changing continuously and the accumulation of this change might be large is introduced…
Periodic solutions of the three body problem are very important for understanding its dynamics either in a theoretical framework or in various applications in celestial mechanics. In this paper we discuss the computation and continuation of…
Randomly-assembled dynamical systems are theoretically predicted to be unstable upon crossing a critical threshold of complexity, as first shown by May. Yet, empirical complex systems exhibit remarkable stability, indicating the presence of…
Stability is a key property of dynamical systems. In some cases, we want to change unstable system into stable one to achieve certain goals in engineering. Here, we present an example of a $3$ dimensional switched system that alternates…
In past works, various schemes for adaptive synchronization of chaotic systems have been proposed. The stability of such schemes is central to their utilization. As an example addressing this issue, we consider a recently proposed adaptive…
This paper is concerned with the study of the stability of dynamical systems evolving on time scales. We first {formalize the notion of matrix measures on time scales, prove some of their key properties and make use of this notion to study…
We consider the relaxation dynamics of two spins coupled to a common bosonic bath. The time evolution is simulated by a generalized master equation derived within a real-time diagrammatic approach. Interference effects due to the coherent…
We analyze the random sequential dynamics of a message passing algorithm for Ising models with random interactions in the large system limit. We derive exact results for the two-time correlation functions and the speed of convergence. The…
We analyze the dynamics of two atoms with a short-ranged pair interaction in a one-dimensional harmonic trap with time-dependent frequency. Our analysis is focused on two representative cases: (i) a sudden change of the trapping frequency…
We study the unitary dynamics of a one-dimensional gas of hard-core bosons trapped into a harmonic potential which varies periodically in time with frequency $\omega(t)$. Such periodic systems can be classified into orbits of different…
We unravel the stationary properties and the interaction quench dynamics of two bosons, confined in a two-dimensional anisotropic harmonic trap. A transcendental equation is derived giving access to the energy spectrum and revealing the…
In this expository and resources chapter we review selected aspects of the mathematics of dynamical systems, stability, and chaos, within a historical framework that draws together two threads of its early development: celestial mechanics…
We obtain sufficient conditions for the stability of the synchronized solutions for a class of coupled dynamical systems. This is accomplished by finding an analytical expression for the transverse Liapunov exponent through spectral…
The stability against perturbations of a dynamical system conserving a generalized phase-space volume is studied by exploiting the similarity between statistical physics formalism and that of ergodic theory. A general continuity theorem is…
We examine the problem of stability of persistent currents in a mixture of two Bose gases trapped in an annular potential. We evaluate the critical coupling for metastability in the transition from quasi-one to two-dimensional motion. We…
Open dynamical systems are mathematical models of machines that take input, change their internal state, and produce output. For example, one may model anything from neurons to robots in this way. Several open dynamical systems can be…
We consider generic differential equations in $\mathbb{R}$ with a finite number of hyperbolic equilibria, which are subject to $\omega$--periodic instantaneous perturbative pulses ($\omega>0$). Using the time-$ \omega$ map of the original…
We study the dynamical response of a harmonically trapped two-component few-fermion mixture to the external gaussian potential barrier moving across the system. The simultaneous role played by inter-particle interactions, rapidity of the…
We study a system of two-mode stochastic oscillators coupled through their collective output. As a function of a relevant parameter four qualitatively distinct regimes of collective behavior are observed. In an extended region of the…
We review recent work on systems with multiple interacting-particles having the dynamical feature of stochastic resetting. The interplay of time scales related to inter-particle interactions and resetting leads to a rich behavior, both…