Related papers: Exact solutions to chaotic and stochastic systems
Stationary solutions to a Fokker-Planck equation corresponding to a noisy logistic equation with correlated Gaussian white noises are constructed. Stationary distributions exist even if the corresponding deterministic system displays an…
Symbolic dynamics has proven to be an invaluable tool in analyzing the mechanisms that lead to unpredictability and random behavior in nonlinear dynamical systems. Surprisingly, a discrete partition of continuous state space can produce a…
The content of this contribution is based on the course on numerical analysis techniques for non-linear dynamics. After introducing basic concepts as the visual analysis of trajectories in phase space and the importance of the nature of…
We show how a simple scheme of symbolic dynamics distinguishes a chaotic from a random time series and how it can be used to detect structural relationships in coupled dynamics. This is relevant for the question at which scale in complex…
We consider a 2-dimensional stochastic differential equation in polar coordinates depending on several parameters. We show that if these parameters belong to a specific regime then the deterministic system explodes in finite time, but the…
The agenda of Dissipative Quantum Chaos is to create a toolbox which would allow us to categorize open quantum systems into "chaotic" and "regular" ones. Two approaches to this categorization have been proposed recently. One of them is…
Nonequilibrium systems driven by additive or multiplicative dichotomous Markov noise appear in a wide variety of physical and mathematical models. We review here some prototypical examples, with an emphasis on {\em analytically-solvable}…
In this paper, we present new results on finite- and fixed-time convergence for dynamical systems using LaSalle-like invariance principles. In particular, we provide first and second-order non-smooth Lyapunov-like results for finite- and…
The predictability problem for systems with different characteristic time scales is investigated. It is shown that even in simple chaotic dynamical systems, the leading Lyapunov exponent is not sufficient to estimate the predictability…
Hyperexponential stability is investigated for dynamical systems with the use of both, explicit and implicit, Lyapunov function methods. A nonlinear hyperexponential control is designed for stabilizing linear systems. The tuning procedure…
The finest state space resolution that can be achieved in a physical dynamical system is limited by the presence of noise. In the weak-noise approximation the neighborhoods of deterministic periodic orbits can be computed as distributions…
Stochastic and coherence resonances appear in nonlinear systems subjected to an external source of noise and are characterized by a maximum response at the optimal value of the noise intensity. This paper shows experimentally that it is…
The current series of papers is concerned with stochastic stability of monotone dynamical systems by identifying the basic dynamical units that can survive in the presence of noise interference. In the first of the series, for the…
An autonomous system of ordinary differential equations describing nonlinear oscillations on the plane is considered. The influence of time-dependent perturbations decaying at infinity in time is investigated. It is assumed that the…
We derive a prediction of dynamical tunneling rates from regular to chaotic phase-space regions combining the direct regular-to-chaotic tunneling mechanism in the quantum regime with an improved resonance-assisted tunneling theory in the…
Deterministic chaotic dynamics presumes that the state space can be partitioned arbitrarily finely. In a physical system, the inevitable presence of some noise sets a finite limit to the finest possible resolution that can be attained. Much…
The control of chaotic systems implies inducing an unpredictable system to follow a desired trajectory using the smallest "force". In low-dimensional continuous systems, one method is that of reconstructing the tangent space, so that the…
We consider the problem of constructing Lyapunov functions for linear differential equations with delays. For such systems it is known that exponential stability implies the existence of a positive Lyapunov function which is quadratic on…
The Lyapunov exponent is well-known in deterministic dynamical systems as a measure for quantifying chaos and detecting coherent regions in physically evolving systems. In this Letter, we show how the Lyapunov exponent can be unified with…
Polynomial chaos is a powerful technique for propagating uncertainty through ordinary and partial differential equations. Random variables are expanded in terms of orthogonal polynomials and differential equations are derived for the…