Dissipative Lattice Systems Research Articles

Emergence of spontaneous symmetry breaking in dissipative lattice systems

A cornerstone of the theory of phase transitions is the observation that many-body systems exhibiting a spontaneous symmetry breaking in the thermodynamic limit generally show extensive fluctuations of an order parameter in large but finite systems. In this work, we introduce the dynamical analog of such a theory. Specifically, we consider local dissipative dynamics preparing an equilibrium steady-state of quantum spins on a lattice exhibiting a discrete or continuous symmetry but with extensive fluctuations in a local order parameter. We show that for all such processes, there exist asymptotically stationary symmetry-breaking states, i.e., states that become stationary in the thermodynamic limit and give a finite value to the order parameter. We give results both for discrete and continuous symmetries and explicitly show how to construct the symmetry-breaking states. Our results show in a simple way that, in large systems, local dissipative dynamics satisfying detailed balance cannot uniquely and efficiently prepare states with extensive fluctuations with respect to local operators. We discuss the implications of our results for quantum simulators and dissipative state preparation.

Discrete breathers in integrate-and-fire oscillator networks

A number of novel dynamical patterns are found in an integrate-and-fire network, including periodic breathers with single, double, and triple wavefronts, and chaotic breathers. It is found that the periodic breathers always propagate whereas the chaotic ones always remain stationary. A novel route to chaotic breathers through intermittency is identified, as opposed to the paths through quasi-periodicity and period-doubling mostly discussed in dissipative lattice systems. The system is found to display prominent bistability, resulting in “hysteresis” loops in single-oscillator inter-spike intervals as the parameter varies.

Asymptotic Behavior of Stochastic Partly Dissipative Lattice Systems in Weighted Spaces

We study stochastic partly dissipative lattice systems with random coupled coefficients and multiplicative/additive white noise in a weighted space of infinite sequences. We first show that these stochastic partly dissipative lattice differential equations generate a random dynamical system. We then establish the existence of a tempered random bounded absorbing set and a global compact random attractor for the associated random dynamical system.

Exponential attractors for lattice dynamical systems in weighted spaces

We first present some sufficient conditions for the existence of exponential attractors for locally coupled lattice dynamical systems in weighted spaces of infinite sequences. Then we apply this result to discuss the existence of exponential attractors for first order lattice systems, partly dissipative lattice systems, and second order lattice systems in weighted spaces of infinite sequences.

FRACTAL DIMENSION OF GLOBAL ATTRACTORS FOR SOME DISSIPATIVE LATTICE SYSTEMS

In this letter, by using a criteria for estimating the fractal dimension of a compact subset of a Hilbert space which is directly obtained from [Chueshov & Lasiecka, 2004], the upper bounds of fractal dimension of the global attractors for first-order and partly dissipative lattice systems, and second-order dissipative lattice systems, are obtained.

Kolmogorov’s ε-Entropy of Bounded Sets in Discrete Spaces and Attractors of Dissipative Lattice Systems

We obtain an estimate of the upper bound for Kolmogorov’s e-entropy for the bounded sets with small “tail” in discrete spaces, then we present a sufficient condition for the existence of a global attractor for dissipative lattice systems in a reflexive Banach discrete space and establish an upper bound of Kolmogorov’s e-entropy of the global attractor for lattice systems.

Attractors and dimension of dissipative lattice systems

In this paper, by using the argument in [Q.F. Ma, S.H. Wang, C.K. Zhong, Necessary and sufficient conditions for the existence of global attractor for semigroup and application, Indiana Univ. Math. J., 51(6) (2002), 1541–1559.], we prove that the condition given in [S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations 200 (2004) 342–368.] for the existence of a global attractor for the semigroup associated with general lattice systems on a discrete Hilbert space is a sufficient and necessary condition. As an application, we consider the existence of a global attractor for a second-order lattice system in a discrete weighted space containing all bounded sequences. Finally, we show that the global attractor for first-order and partly dissipative lattice systems corresponding to (partly dissipative) reaction–diffusion equations and second-order dissipative lattice systems corresponding to the strongly damped wave equations have finite fractal dimension if the derivative of the nonlinear term is small at the origin.

Spatial structures in dissipative media at the threshold of chaos

Multistable states of dissipative lattice systems at the threshold of dynamic chaos are investigated. A classification is developed for possible spatial structures and the mechanisms of their coexistence and formation are examined.