Abstract:
The paper provides a new integral formula for the largest Lyapunov exponent of Gaussian matrices, which is valid in the real, complex and quaternion-valued cases. This formula is applied to derive asymptotic expressions for the largest Lyapunov exponent when the size of the matrix is large and compare the Lyapunov exponents in models with a spike and no spikes.

Abstract:
We present a geometric argument that explains why some systems having vanishing largest Lyapunov exponent have underlying dynamics aspects of which can be effectively described by the Tsallis entropy. We rely on a comparison of the generalised additivity of the Tsallis entropy versus the ordinary additivity of the BGS entropy. We translate this comparison in metric terms by using an effective hyperbolic metric on the configuration/phase space for the Tsallis entropy versus the Euclidean one in the case of the BGS entropy. Solving the Jacobi equation for such hyperbolic metrics effectively sets the largest Lyapunov exponent computed with respect to the corresponding Euclidean metric to zero. This conclusion is in agreement with all currently known results about systems that have a simple asymptotic behaviour and are described by the Tsallis entropy.

Abstract:
Emergent behaviors of collective intelligence systems, exemplified by swarm model, have attracted broad interests in recent years. However, current research mostly stops at observational interpretations and qualitative descriptions of emergent phenomena and is essentially short of quantitative analysis and evaluation. In this paper, we conduct a quantitative study on the emergence of swarm model by using chaos analysis of complex dynamic systems. This helps to achieve a more exact understanding of emergent phenomena. In particular, we evaluate the emergent behaviors of swarm model quantitatively by using the chaos and stability analysis of swarm model based on largest Lyapunov exponent. It is concluded that swarm model is at the edge of chaos when emergence occurs, and whether chaotic or stable at the beginning, swarm model will converge to stability with the elapse of time along with interactions among agents. 1. Introduction Collective intelligence brings up a bottom-up approach, which is essentially different from the top-down one as in conventional systems. The bottom-up approach incites complex global behaviors through local interactions among agents [1]. Collective intelligence provides an effective approach to solving complex problems including adaptation to dynamic environments. Prediction and control are two main concerns with emergent behaviors. In order to predict emergent behaviors, the first thing is analysis and evaluation of dynamic behaviors of systems. Wolf et al. presented an “equation-free” method to analyze and evaluate the trends of systems [2]. Zhu presented a formal theory, called Scenario Calculus, to reason about the emergent behaviors of multiagent systems [3]. Gazi and Passino simulated and observed the characteristics of behaviors under different conditions to analyze the stability of social foraging swarm [4]. Pedrami and Gordon used an additional controllable variable to study the changes of interior energy of swarm system [5]. On the control of emergent behaviors, chaos analysis was employed to study the behaviors of swarming. Qu et al. utilized nonlinear chaos time series to control emergent behaviors [6]. Ishiguro et al. presented an immunological approach to controlling the behavior of self-organized robots [7, 8]. Meng et al. presented an algorithm integrating ant colony optimization and particle swarm optimization to build distributed multiagent systems [9]. Waltman and Kaymak studied multiagent Q-learning for deciding how to behave in an unknown environment [10]. Generally speaking, current research mainly uses

Abstract:
Dynamics of driven Frenkel-Kontorova model is examined by using largest Lyapunov exponent computational technique. Obtained results show that comparing to the usual way where behavior of the system in the presence of external forces is examined by analyzing its dynamical response function, the largest Lyapunov exponent analysis often represents a better tool to estimate the system dynamics. In the dc driven system, the critical depinning force that needs to be applied on particular structure could be estimated according to the largest Lyapunov exponent. In the dc+ac driven system, calculation of the largest Lyapunov exponent not only offers the most accurate way to detect the presence of Shapiro steps but completely reflects the amplitude dependence of the step size and the critical depinning force. The largest Lyapunov exponent as a function of the ac amplitude calculated in the pinned regime represents a mirror image of the amplitude dependence of the critical depinning force obtained from the dynamical response function when both forces are applied.

Abstract:
We calculate analytically the largest Lyapunov exponent of the so-called $\alpha XY$ Hamiltonian in the high energy regime. This system consists of a $d$-dimensional lattice of classical spins with interactions that decay with distance following a power-law, the range being adjustable. In disordered regimes the Lyapunov exponent can be easily estimated by means of the "stochastic approach", a theoretical scheme based on van Kampen's cumulant expansion. The stochastic approach expresses the Lyapunov exponent as a function of a few statistical properties of the Hessian matrix of the interaction that can be calculated as suitable microcanonical averages. We have verified that there is a very good agreement between theory and numerical simulations.

Abstract:
A general indicator of the presence of chaos in a dynamical system is the largest Lyapunov exponent. This quantity provides a measure of the mean exponential rate of divergence of nearby orbits. In this paper, we show that the so-called two-particle method introduced by Benettin et al. could lead to spurious estimations of the largest Lyapunov exponent. As a comparator method, the maximum Lyapunov exponent is computed from the solution of the variational equations of the system. We show that the incorrect estimation of the largest Lyapunov exponent is based on the setting of the renormalization time and the initial distance between trajectories. Unlike previously published works, we here present three criteria that could help to determine correctly these parameters so that the maximum Lyapunov exponent is close to the expected value. The results have been tested with four well known dynamical systems: Ueda, Duffing, R\"ossler and Lorenz.

Abstract:
We propose an analog interference method for measuring the largest Lyapunov exponent for the optical fields generated by scattering objects and media. The method is further developed to make a device for high-speed real-time measurements of transverse correlation function of the optical fields.

Abstract:
We present a general formalism for computing the largest Lyapunov exponent and its fluctuations in spatially extended systems described by diffusive fluctuating hydrodynamics, thus extending the concepts of dynamical system theory to a broad range of non-equilibrium systems. Our analytical results compare favourably with simulations of a lattice model of heat conduction. We further show how the computation of the Lyapunov exponent for the Symmetric Simple Exclusion Process relates to damage spreading and to a two-species pair annihilation process, for which our formalism yields new finite size results.

Abstract:
The largest Lyapunov exponent of an ergodic Hamiltonian system is the rate of exponential growth of the norm of a typical vector in the tangent space. For an N-particle Hamiltonian system, with a smooth Hamiltonian of the type p^2 + v(q), the evolution of tangent vectors is governed by the Hessian matrix V of the potential. Ergodicity implies that the Lyapunov exponent is independent of initial conditions on the energy shell, which can then be chosen randomly according to the microcanonical distribution. In this way a stochastic process V(t) is defined, and the evolution equation for tangent vectors can now be seen as a stochastic differential equation. An equation for the evolution of the average squared norm of a tangent vector can be obtained using the standard theory in which the average propagator is written as a cummulant expansion. We show that if cummulants higher than the second one are discarded, the Lyapunov exponent can be obtained by diagonalizing a small-dimension matrix, which, in some cases, can be as small as 3x3. In all cases the matrix elements of the propagator are expressed in terms of correlation functions of the stochastic process. We discuss the connection between our approach and an alternative theory, the so-called geometric method.

Abstract:
Echo State Networks are efficient time-series predictors, which highly depend on the value of the spectral radius of the reservoir connectivity matrix. Based on recent results on the mean field theory of driven random recurrent neural networks, enabling the computation of the largest Lyapunov exponent of an ESN, we develop a cheap algorithm to establish a local and operational version of the Echo State Property.