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.

A hysteretic model for rainfall-runoff of a simplifed catchment

<div>The soil can be modelled as a porous medium in which the three phases of matter coexist and produce the emergent phenomenon of hysteresis.</div><div>Rate-independent hysteresis is a nonlinear phenomenon where the output depends not only on the current input but also the previous history of inputs to the system. In multiphase porous media such as soils, the hysteresis is in the relationship between the soil-moisture content, and the capillary pressure.</div><div>In this work, we develop a simplified hysteretic rainfall-runoff model consisting of the following subsystems that capture much of the physics of flow through a slab of soil:</div><div>1) A slab of soil where rainfall enters and if enough water is present in the soil, it will subsequently drain into the groundwater reservoir. This part of the model is represent by ODE with a Preisach operator.</div><div>2) A runoff component: If the rainfall exceeds the maximum infiltration rate of the soil, the excess will become surface runoff. This part of the model is represented by a series of two hysteretic reservoirs instead of the two linear reservoirs in the literature.</div><div>3) A ground water storage and outflow subsystem component: this is also modelled by a hysteretic reservoir. Finally, the outputs from the groundwater output and the overland flow are combined to give the total runoff. We will examine this model and compare it with non-hysteretic case both qualitatively and quantitively.</div>

Periodic solutions of a hysteresis model for breathing

We propose to model the lungs as a viscoelastic deformable porous medium with a hysteretic pressure–volume relationship described by the Preisach operator. Breathing is represented as an isothermal time-periodic process with gas exchange between the interior and exterior of the body. The main result consists in proving the existence of a periodic solution under an arbitrary periodic forcing in suitable function spaces.