A New Way to Compute the Lyapunov Characteristic Exponents for Non-Smooth and Discontinues Dynamical Systems

One of the most important problems of nonlinear dynamics is related to the development of methods concerning the identification of the dynamical modes of the corresponding systems. The classical method is related to the calculation of the Lyapunov characteristic exponents ( LCEs ). Usually, to implement the classical algorithms for the LCEs calculation the smoothness of the right-hand sides of the corresponding equations is required. In this work, we propose a new algorithm for the LCEs computation in systems with strong nonlinearities (these nonlinearities can not be linearized ) including the hysteresis. This algorithm uses the values of the Jacobi matrix in the vicinity of singularities of the right-hand sides of the corresponding equations. The proposed modification of the algorithm is also can be used for systems containing such design hysteresis nonlinearity as the Preisach operator (thus, this modification can be used for all members of the hysteresis family). Moreover, the proposed algorithm can be successfully applied to the well-known chaotic systems with smooth nonlinearities . Examples of dynamical systems with hysteresis nonlinearities , such as the Lorentz system with hysteresis friction and the van der Pol oscillator with hysteresis block, are considered. These examples illustrate the efficiency of the proposed algorithm.

Vibrations in Single-Degrees-of-Freedom Sampled-Data Linear Mechanical Systems

This paper aims to present that the effect of sampling can result in multi-frequency vibration even in the case of a single-degree-of-freedom linear mechanical model. Even though the sampled-data systems have an infinite number of characteristic exponents due to sampling, the vibrations of these systems can still be characterized by an effective system model with a single dominant frequency. However, as this paper shows, additional harmonics become relevant, resulting in multi-frequency vibrations depending on the magnitude of applied control parameters. The vibrations presented by the time histories of vibration and their spectra resulted in numerical simulation of the sampled-data system.

An Improved Chaotic Detection System for Metal Detectors

This paper first applies a chaotic system-Duffing oscillator to a metal detector, and uses RHR algorithm to compute two Lyapunov characteristics exponents of the Duffing system. In this way, the two Lyapunov characteristic exponents can help to judge the Duffing system being chaotic or not quantitatively. And also help to get the threshold value more accurately. Then a simulation model of Duffing system fit for detectors is established by Matlab. Simulation results indicate that an suitable Duffing system can improve the sensitivity of a detector effectively.

OPTION IMPLIED VIX, SKEW AND KURTOSIS TERM STRUCTURES

Comparisons are made of the Chicago Board of Options Exchange (CBOE) skew index with those derived from parametric skews of bilateral gamma models and from the differentiation of option implied characteristic exponents. Discrepancies can be due to strike discretization in evaluating prices of powered returns. The remedy suggested employs a finer and wider set of strikes obtaining additional option prices by interpolation and extrapolation of implied volatilities. Procedures of replicating powered return claims are applied to the fourth power and the derivation of kurtosis term structures. Regressions of log skewness and log excess kurtosis on log maturity confirm the positivity of decay in these higher moments. The decay rates are below those required by processes of independent and identically distributed increments.

A Re-Examination of Various Resonances in Parametrically Excited Systems1

The dynamics of parametrically excited systems are characterized by distinct types of resonances including parametric, combination, and internal. Existing resonance conditions for these instability phenomena involve natural frequencies and thus are valid when the amplitude of the parametric excitation term is zero or close to zero. In this paper, various types of resonances in parametrically excited systems are revisited and new resonance conditions are developed such that the new conditions are valid in the entire parametric space, unlike existing conditions. This is achieved by expressing resonance conditions in terms of “true characteristic exponents” which are defined using characteristic exponents and their non-uniqueness property. Since different types of resonances may arise depending upon the class of parametrically excited systems, the present study has categorized such systems into four classes: linear systems with parametric excitation, linear systems with combined parametric and external excitations, nonlinear systems with parametric excitation, and nonlinear systems with combined parametric and external excitations. Each class is investigated separately for different types of resonances, and examples are provided to establish the proof of concept. Resonances in linear systems with parametric excitation are examined using the Lyapunov–Poincaré theorem, whereas Lyapunov–Floquet transformation is utilized to generate a resonance condition for linear systems with combined excitations. In the case of nonlinear parametrically excited systems, nonlinear techniques such as “time-dependent normal forms” and “order reduction using invariant manifolds” are employed to express various resonance conditions. It is found that the forms of new resonance conditions obtained in terms of ‘true characteristic exponents’ are similar to the forms of existing resonance conditions that involve natural frequencies.

Stability Analysis Of Delayed Fractional Integro-Differential Equations With Applications Of RLC Circuits

This article presents the stability analysis of delay integro-differentialequations with fractional order derivative via some approximation techniques forthe derived nonlinear terms of characteristic exponents. Based on these techniques,the existence of some analytical solutions at the neighborhood of their equilibriumpoints is proved. Stability charts are constructed and so both of the critical timedelay and critical frequency formulae are obtained. The impact of this work into thegeneral RLC circuit applications exposing the delay and fractional order derivativesis discussed.

Lyapunov exponents for expansive homeomorphisms

We address the problem of defining Lyapunov exponents for an expansive homeomorphism f on a compact metric space (X, dist) using similar techniques as those developed in Barreira and Silva [Lyapunov exponents for continuous transformations and dimension theory, Discrete Contin. Dynam. Sys.13 (2005), 469–490]; Kifer [Characteristic exponents of dynamical systems in metric spaces, Ergod. Th. Dynam. Sys.3 (1983), 119–127]. Under certain conditions on the topology of the space X where f acts we obtain that there is a metric D defining the topology of X such that the Lyapunov exponents of f are different from zero with respect to D for every point x ∈ X. We give an example showing that this may not be true with respect to the original metric dist. But expansiveness of f ensures that Lyapunov exponents do not vanish on a Gδ subset of X with respect to any metric defining the topology of X. We define Lyapunov exponents on compact invariant sets of Peano spaces and prove that if the maximal exponent on the compact set is negative then the compact is an attractor.