The dominant Lyapunov exponent of a dynamical system measures the average rate at which nearby trajectories of a system diverge. Even though a positive exponent provides evidence for chaotic dynamics and upredictability, there may predictability of the time series over some finite time periods. In this paper one version of a local Lyapunov exponent is defined for a dynamic system perturbed by n...

BACKGROUND The role of ischemia reperfusion contributing to functional impairment in lower extremity peripheral arterial disease (PAD) patients has not previously been elucidated. The evaluation of gait variability patterns has proven useful in many pathologic populations. Therefore, the purpose of this study is to isolate and determine the specific effect of the acute reperfusion phase of isch...

The presence of chaos in walking is considered to provide a stable, yet adaptable means for locomotion. This study examined whether lower limb amputation and subsequent prosthetic rehabilitation resulted in a loss of complexity in amputee gait. Twenty-eight individuals with transtibial amputation participated in a 6 week, randomized cross-over design study in which they underwent a 3 week adapt...

We study the random Fibonacci sequences defined by F 1 = F 2 = e F 1 = e F 2 = 1 and for n ≥ 1, F n+2 = F n+1 ± Fn (linear case) and e F n+2 = | e F n+1 ± e Fn| (non-linear case), where each ± sign is independent and either + with probability p or − with probability 1 − p (0 < p ≤ 1). Our main result is that the exponential growth of Fn for 0 < p ≤ 1, and of e Fn for 1/3 ≤ p ≤ 1 is almost surel...

We applied nonlinear and chaos analysis to fingertip pulse wave data. The largest Lyapunov exponent, a measure of the “divergence” of the trajectory of the attractor in phase space, was found to be a useful index of mental health in humans, particularly for the early detection of dementia and depressive psychosis, and for monitoring mental changes in healthy persons. Most of the methods used fo...

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,...

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...

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 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...