Stability of Periodic Solutions for Hysteresis-Delay Differential Equations

Abstract

We study an interplay between delay and discontinuous hysteresis in dynamical systems. After having established existence and uniqueness of solutions, we focus on the analysis of stability of periodic solutions. The main object we study is a Poincaré map that is infinite-dimensional due to delay and non-differentiable due to hysteresis. We propose a general functional framework based on the fractional order Sobolev-Slobodeckij spaces and explicitly obtain a formal linearization of the Poincaré map in these spaces. Furthermore, we prove that the spectrum of this formal linearization determines the stability of the periodic solution and then reduce the spectral analysis to an equivalent finite-dimensional problem.

Optimal control of a population dynamics model with hysteresis

“…[1]). During the last three decades, hysteresis operators have been extensively applied in modeling, analysis, and control of a variety of irreversible nonlinear phenomena in applied sciences including phase transitions [2][3][4][5][6][7], porous media flow [8][9][10][11][12], thermostat models [13][14][15][16][17], concrete carbonation [18][19][20][21][22] and many others. In the same vein, system (1.1)-(1.5) (without the control u) was introduced in [23] (see, also, [24,25]) to model the evolution of populations in the vegetation-preypredator framework when diffusive effects in the dynamics of three species are taken into account and the food density for the prey exhibits hysteretic character.…”