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Lyapunov exponents estimation for hysteretic systems

机译：滞后系统的Lyapunov指数估计

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This article discusses the Lyapunov exponent estimation of non-linear hysteretic systems by adapting the classical algorithm by Wolf and co-workers [Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A., 1985. Determining Lyapunov exponents from a times series. Physica D 16, 285-317.1. This algorithm evaluates the divergence of nearby orbits by monitoring a reference trajectory, evaluated from the equations of motion of the original hysteretic system, and a perturbed trajectory resulting from the integration of the linearized equations of motion. The main issue of using this algorithm for non-linear, rate-independent, hysteretic systems is related to the procedure of linearization of the equations of motion. The present work establishes a procedure of linearization performing a state space split and assuming an equivalent viscous damping in order to represent hysteretic dissipation in the linearized system. The dynamical response of a single-degree of freedom pseudoelastic shape mem_ory alloy (SMA) oscillator is discussed as an application of the proposed algorithm. The restitution force of the oscillator is provided by an SMA element described by a rate-independent, hysteretic, thermomechanical constitutive model. Two different modeling cases are considered for isothermal and non-isothermal heat transfer conditions, and numerical simulations are performed for both cases. The evaluation of the Lyapunov expo_nents shows that the proposed procedure is capable of quantifying chaos capturing the non-linear dissipation of hysteretic systems.
机译：本文通过采用Wolf和他的同事们的经典算法[Wolf，A.，Swift，JB，Swinney，HL，Vastano，JA，1985，讨论了非线性滞后系统的Lyapunov指数估计。从一个时间确定Lyapunov指数系列。 Physica D 16，285-317.1。该算法通过监视参考轨迹（从原始磁滞系统的运动方程评估）和由线性化运动方程的积分产生的扰动轨迹，来评估附近轨道的发散。将这种算法用于非线性，与速率无关的磁滞系统的主要问题与运动方程的线性化过程有关。本工作建立了执行状态空间分割并假设等效粘性阻尼的线性化过程，以表示线性化系统中的滞后耗散。讨论了单自由度拟弹性形状记忆合金（SMA）振荡器的动力学响应，并将其作为该算法的应用。振荡器的恢复力由SMA元件提供，该SMA元件与速率无关，迟滞，热机械本构模型描述。对于等温和非等温传热条件，考虑了两种不同的建模案例，并针对这两种案例进行了数值模拟。对Lyapunov指数的评估表明，所提出的过程能够量化捕获滞后系统非线性耗散的混沌。

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