The vibration acceleration time history of the cutter holder was separated into three parts; namely, chatter free, transition and chatter processes. The reconstructed attractor and probability distribution of vibration acceleration time series were studied in order to observe the system’s behavior. The Lyapunov exponent andKolmogorov entropy were used to help judge the cutting state. Meanwhile,...

We study the largest Lyapunov exponent of the response of a two dimensional non-Hamiltonian system driven by additive white noise. The specific system we consider is the third-order truncated normal form of the unfolding of a Hopf bifurcation. We show that in the small-noise limit the top Lyapunov exponent always approaches zero from below (and is thus negative for noise sufficiently small); we...

A one-dimensional chain of sporadic maps with asymmetric nearest neighbour couplings is numerically studied. It is shown that in the region of strong asymmetry the system becomes spatially fully synchronized, even in the thermodinamic limit, while the Lyapunov exponent is zero. For weak asymmetry the synchronization is no more complete, and the Lyapunov exponent becomes positive. In addition on...

We present a route to high-dimensional chaos, that is, chaos with more than one positive Lyapunov exponent. In this route, as a system parameter changes, a subsystem becomes chaotic through, say, a cascade of period-doubling bifurcations, after which the complementary subsystem becomes chaotic, leading to an additional positive Lyapunov exponent for the whole system. A characteristic feature of...

This paper investigates the relation between chaos and swarm intelligence. The swarm intelligent model is represented as an iterated function system (IFS). The dynamic trajectory of the particle is sensitive on the parameter values of IFS. The Lyapunov exponent and the correlation dimension were calculated and analyzed. Our preliminary research results illustrate that the performance of the swa...

We calculate analytically the largest Lyapunov exponent of the so-called αXY 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 powerlaw, the range being adjustable. In disordered regimes the Lyapunov exponent can be easily estimated by means of the “stochastic approach”, a theoretical ...

We consider one-step cocycles of 2ˆ 2 matrices, and we are interested in their Lyapunov-optimizing measures, i.e., invariant probability measures that maximize or minimize a Lyapunov exponent. If the cocycle is dominated, that is, the two Lyapunov exponents are uniformly separated along all orbits, then Lyapunov-optimizing measures always exist, and are characterized by their support. Under an ...

We investigate the dynamical origin for the occurrence of asynchronous hyperchaos and chaos via blowout bifurcations in coupled chaotic systems. An asynchronous hyperchaotic or chaotic attractor with a positive or negative second Lyapunov exponent appears through a blowout bifurcation. It is found that the sign of the second Lyapunov exponent of the newly born asynchronous attractor, exhibiting...

The problem of evaluation of Lyapunov exponent in queueing network analysis is considered based on models and methods of idempotent algebra. General existence conditions for Lyapunov exponent to exist in generalized linear stochastic dynamic systems are given, and examples of evaluation of the exponent for systems with matrices of particular types are presented. A method which allow one to get ...