Entropy in Dynamic Systems

In order to measure and quantify the complex behavior of real-world systems, either novel mathematical approaches or modifications of classical ones are required to precisely predict, monitor, and control complicated chaotic and stochastic processes. Though the term of entropy comes from Greek and emphasizes its analogy to energy, today, it has wandered to different branches of pure and applied sciences and is understood in a rather rough way, with emphasis placed on the transition from regular to chaotic states, stochastic and deterministic disorder, and uniform and non-uniform distribution or decay of diversity. This collection of papers addresses the notion of entropy in a very broad sense. The presented manuscripts follow from different branches of mathematical/physical sciences, natural/social sciences, and engineering-oriented sciences with emphasis placed on the complexity of dynamical systems. Topics like timing chaos and spatiotemporal chaos, bifurcation, synchronization and anti-synchronization, stability, lumped mass and continuous mechanical systems modeling, novel nonlinear phenomena, and resonances are discussed

A novel dry friction modeling and its impact on differential equations computation and lyapunov exponents estimation

A novel dry friction modeling and its impact on differential equations computation and Lyapunov exponents estimation are studied in this paper. A brief review of some existing standard friction laws are presented and novel continuous friction model is proposed, which takes into account some elements of the mentioned friction models. We show that our continuous friction model is suitable for analysis of stick-slip vibrations caused by dry friction and is more efficient from a computational point of view in comparison with the other presented friction models. Its advantages are illustrated and discussed using a two degree-of-freedom model. Although there are numerous works in the scientific literature dedicated to stick-slip vibrations, we consider a rigid body lying on a belt which moves at non-constant velocity that is less investigated. The behavior of the system is monitored via standard motion analysis. Time series, phase portraits, bifurcation diagram as well as the Lyapunov exponents are reporte

Computational approach for complete Lyapunov functions

Ordinary differential equations arise in a variety of applications, including climate modeling, electronics, predator-prey modeling, etc., and they can exhibit highly complicated dynamical behaviour. Complete Lyapunov functions capture this behaviour by dividing the phase space into two disjoint sets: the chain-recurrent part and the transient part. If a complete Lyapunov function is known for a dynamical system the qualitative behaviour of the system’s solutions is transparent to a large degree. The computation of a complete Lyapunov function for a given system is, however, a very hard task. We present significant improvements of an algorithm recently suggested by the authors to compute complete Lyapunov functions. Previously this methodology was incapable to fully detect chain-recurrent sets in dynamical systems with high differences in speed. In the new approach we replace the system under consideration with another one having the same solution trajectories but such that they are traversed at a more uniform speed. The qualitative properties of the new system such as attractors and repellers are the same as for the original one. This approach gives a better approximation to the chain-recurrent set of the system under study

Simple model of bouncing ball dynamics. Displacement of the limiter assumed as a cubic function of time

Nonlinear dynamics of a bouncing ball moving vertically in a gravitational field and colliding with a moving limiter is considered and the Poincare map, describing evolution from an impact to the next impact, is described. Displacement of the limiter is assumed as periodic, cubic function of time. Due to simplicity of this function analytical computations are possible. Several dynamical modes, such as fixed points, 2 - cycles and chaotic bands are studied analytically and numerically. It is shown that chaotic bands are created from fixed points after first period doubling in a corner-type bifurcation. Equation for the time of the next impact is solved exactly for the case of two subsequent impacts occurring in the same period of limiter's motion making analysis of chattering possible.Comment: 8 pages, 1 figure, presented at the DSTA 2011 conference, Lodz, Polan

Influence of geometric and physical nonlinearities on the internal resonances of a finite continuous rod with a microstructure

In this work, nonlinear longitudinal vibrations of a finite composite rod are studied including geometric and physical nonlinearities. An original boundary value problem for a heterogeneous rod yielded by the macroscopic approximation obtained earlier by the higher-order asymptotic homogenization method is used. The effects of internal resonances and modes coupling are predicted, validated and analyzed. The defined novel continuous problem governed by PDEs is solved using space-discretization and the method of multiple time scales. We are aimed at understanding and analyzing how the presence of the microstructure influences the processes of mode interaction. It is shown that, depending on a scaling relation between the amplitude of the vibrations and the size of the unit cell, different scenarios of the modes coupling can be realized. Additionally to the asymptotic solution, numerical simulation of the modes coupling is performed by means of the Runge-Kutta fourth-order method. The obtained numerical and analytical results demonstrate good qualitative agreement

Vibration suppression and angle tracking of a fire-rescue ladder

This paper mainly considers vibration suppression and angle tracking of a fire-rescue ladder system. The dynamical model is regarded as a segmented Euler–Bernoulli beam with gravity and tip mass, described by a set of motion equations and boundary conditions. Based on the nonlinear Euler–Bernoulli beam model, two active boundary controllers are proposed to achieve the control objectives. The elastic deflection and the angular error in the closed-loop system are proven to converge exponentially to a small neighborhood of zero. Numerical simulations based on finite difference method verify the effectiveness and the ascendancy of active boundary controllers