Differential Equations with a Small Parameter and Multipeak Oscillations

Abstract

In this paper, we study a nonlinear dynamical system of autonomous ordinary differential equations with a small parameter $\mu$ such that two variables $x$ and $y$ are fast and another one $z$ is slow. If we take the limit as $\mu \to 0$, then this becomes a “degenerate system” included in the one-parameter family of two-dimensional subsystems of fast motions with the parameter $z$ in some interval. It is assumed that in each subsystem there exists a structurally stable limit cycle $l_z$. In addition, in the complete dynamical system there is some structurally stable periodic orbit $L$ that tends to a limit cycle $l_{z_0}$ for some $z=z_0$ as $\mu$ tends to zero. We can define the first return map, or the Poincaré map, on a local cross section in the hyperplane $(y,z)$ orthogonal to $L$ at some point. We prove that the Poincaré map has an invariant manifold for the fixed point corresponding to the periodic orbit $L$ on a guaranteed interval over the variable $y$, and the interval length is separated from zero as $\mu$ tends to zero. The proved theorem allows one to formulate some sufficient conditions for the existence and/or absence of multipeak oscillations in the complete dynamical system. As an example of application of the obtained results, we consider some kinetic model of the catalytic reaction of hydrogen oxidation on nickel.