Evaluation Of Lyapunov Exponents Research Articles

Investigation of the hyperchaos and control in the fractional order financial system with profit margin

This research study proposes a novel hyperchaotic finance system with profit margin and then utilizes the Adomian Decomposition Method (ADM) to tackle the solution of the associated Caputo-derivative based fractional-order hyperchaotic finance system with profit margin. Numerical simulations and analyses, such as the evaluation of Lyapunov exponents (LE), the creation of bifurcation diagrams (BD), the complexity analysis (CA), and the 0-1 test, are used to get a full picture of the system. The results of our study demonstrate the occurrence of periodic, chaotic, and hyperchaotic dynamics in the system. Furthermore, the overall criteria for the existence and the uniqueness of the exact solutions for Caputo fractional-order models are presented. In addition, we present a control methodology for the fractional-order hyperchaotic financial system utilizing the Laplace transform and linear feedback control. Significantly, our research showcases a noteworthy correlation between the analytical results and numerical simulations, emphasizing the soundness and effectiveness of our methodology.

Topaj–Pikovsky Involution in the Hamiltonian Lattice of Locally Coupled Oscillators

We discuss the Hamiltonian model of an oscillator lattice with local coupling. The Hamiltonian model describes localized spatial modes of nonlinear the Schrodinger equation with periodic tilted potential. The Hamiltonian system manifests reversibility of the Topaj–Pikovsky phase oscillator lattice. Furthermore, the Hamiltonian system has invariant manifolds with asymptotic dynamics exactly equivalent to the Topaj–Pikovsky model. We examine the stability of trajectories belonging to invariant manifolds by means of numerical evaluation of Lyapunov exponents. We show that there is no contradiction between asymptotic dynamics on invariant manifolds and conservation of phase volume of the Hamiltonian system. We demonstrate the complexity of dynamics with results of numerical simulations.

Lyapunov exponents and the Belousov-Zhabotinsky oscillator: an interactive computational approach

The Belousov-Zhabotinsky (BZ) chemical oscillator is the most studied oscillator. It has been modelled on the basis of single-step mechanisms which have been continuously refined since the seminal manuscript by Field, Koros and Noyes in 1972. This manuscript reports on a unique way of modelling the global dynamics of the oscillator by assuming that the BZ oscillator has shown chaotic behaviour. The unique mathematical definition of chaos is very stringent, and, in this manuscript, we attempt to trace this unique exotic behaviour by the use of ‘onto’ maps of the interval onto itself which are known to exhaustively show a universal sequence of states that has all the hallmarks of chaotic behaviour. A series of one-humped maps of the interval display, through iterations and subsequent symbolic dynamics, a universal sequence of steps that commence with period-doubling, culminating in chaotic behaviour at some accumulation point of an appropriate bifurcation parameter.We put this theory to the test for the BZ oscillator in this manuscript by selecting a unique continuous map of the interval. This was then decomposed by an iterative treatment. Metric entropy and subsequent arbiter of chaotic behaviour was determined by evaluation of Lyapunov exponents which were then compared to observed BZ oscillator states. Our proposed map satisfactorily modelled the global dynamics of the BZ oscillator; predicted period-doubling, and a regime after a critical bifurcation parameter, where chaotic sequences were dense.We also produce, in the Addendum, an iterative MatLab procedure that any reader can utilize to reveal the type of states and behaviour reported here.

Macro- and micro-chaotic structures in the Hindmarsh-Rose model of bursting neurons.

We study a plethora of chaotic phenomena in the Hindmarsh-Rose neuron model with the use of several computational techniques including the bifurcation parameter continuation, spike-quantification, and evaluation of Lyapunov exponents in bi-parameter diagrams. Such an aggregated approach allows for detecting regions of simple and chaotic dynamics, and demarcating borderlines-exact bifurcation curves. We demonstrate how the organizing centers-points corresponding to codimension-two homoclinic bifurcations-along with fold and period-doubling bifurcation curves structure the biparametric plane, thus forming macro-chaotic regions of onion bulb shapes and revealing spike-adding cascades that generate micro-chaotic structures due to the hysteresis.

Evaluation of Lyapunov exponents and scaling functions from time series

An algorithm to calculate the Lyapunov exponents from an experimental time series is discussed and tested on simulated data of various nonlinear systems. It is applied to the analysis of experiments on a parametrically modulated nuclear magnetic resonance laser system, and the results are compared with the values of the dimensions and entropies determined by different methods. The distribution of the Lyapunov exponents and the corresponding scaling function is discussed.