Differential Equations with Bistable Nonlinearity

Abstract

We study bounded solutions of differential equations with bistable nonlinearity by numerical and analytic methods. A simple mechanical model of circular pendulum with magnetic suspension in the upper equilibrium position is regarded as a bistable dynamical system simulating a supersensitive seismograph. We consider autonomous differential equations of the second and fourth orders with discontinuous piecewise linear and cubic nonlinearities. Bounded solutions with finitely many zeros, including solitonlike solutions with two zeros and kinklike solutions with several zeros are studied in detail. It is shown that, to within the sign and translation, the bounded solutions of the analyzed equations are uniquely determined by the integer numbers $$ n=\left[\frac{d}{l}\right] $$ where d is the distance between the roots of these solutions and l is a constant characterizing the intensity of nonlinearity. The existence of bounded chaotic solutions is established and the exact value of space entropy is found for periodic solutions.