Abstract
In this paper, we study a nonlinear dynamical system of autonomous ordinary differentialequations 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 “degeneratesystem” included in the one-parameter family of two-dimensional subsystems offast motions with the parameter \(z\) in some interval. It is assumed that in each subsystem there existsa structurally stable limit cycle \(l_z\) . In addition, in the completedynamical 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 Poincarémap, on a local cross section in the hyperplane \((y,z)\) orthogonal to \(L\) at some point. We prove that the Poincaré map has an invariantmanifold 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 sufficientconditions for the existence and/or absence of multipeak oscillations in the complete dynamicalsystem. As an example of application of the obtained results, we consider some kinetic model ofthe catalytic reaction of hydrogen oxidation on nickel.
