Coupled van der Pol oscillators containing a fifth-order conductance characteristic are analyzed using a method due to Endo and Mori. This structure has been proposed for modeling the myoelectrical activity of the human large intestine since it produces stable conditions of zero, two single-mode frequencies, and also a double-mode condition. The paper first considers the case of oscilators having equal frequencies and it is shown that for a certain range of parameters double modes are possible for two oscillators, which is in contrast to the case of conventional van der Pol dynamics. The theoretical results have been experimentally verified using both analog simulations and electronic circuitry. In digestive tract modeling, the coupled oscillators must often have unequal intrinsic frequencies and this case is also considered and shown to give similar but more complex parameter conditions for the establishment of double modes.

Multimode oscillations in mutually coupled van der Pol type oscillators with fifth-power nonlinear characteristics

Structural stability on manifolds with boundary

This paper is a self-contained exposition of the theory of averaging for periodic and quasiperiodic systems, with the emphasis being on the author’s research (part of it joint work with Clark Robinson) on qualitative aspects of nonlinear resonance. Many topics in averaging theory are not covered, among them: averaging for systems more general than quasiperiodic; relations between averaging and multiple time-scale methods; Eckhaus’s approach to averaging; combinations of averaging with matching of asymptotic expansions. The principal question which is addressed is: when does averaging (to first or higher order) lead to an accurate qualitative description of the solutions of the original (unaveraged) equation? By qualitative description we mean both locally (existence and stability of certain invariant sets such as periodic orbits, or almost invariant ‘lingering’ and globally (connecting orbits between invariant sets, or claims that in certain large regions all orbits drift in a certain direction).

Qualitative Theory of Nonlinear Resonance by Averaging and Dynamical Systems Methods

On nonconservative periodic systems close to two-dimensional hamiltonian☆

This paper continues the mathematical study of spin/orbit coupling which was begun in earlier articles by the second author. The equation studied is\(\ddot \theta = \varepsilon P(t,\theta ) + \varepsilon T(t,\dot \theta )\) whereP andT are periodic int and the angle θ. No specific form forP andT is assumed but only the conditions necessary to analyze the system near resonance or a ground state. The behavior of an averaged system of these equations is shown to be reflected qualitatively in the actual equations by rigorous mathematical methods.

Some mathematical aspects of spin-orbit resonance. II

Resonance capture in certain nearly Hamiltonian systems

Perturbed systems of differential equations are studied which are close to Hamiltonian systems of the type for which small divisor problems arise. If the non-Hamiltonian terms involve both damping and self-excitation, behavior characteristic of Hamiltonian systems can occur but is confined to small regions of space in which these opposite effects are in balance. The results provide a partial justification for truncating the Fourier series occurring in these problems. The principle method is local averaging, which is used to obtain global results by compactness arguments.

Nearly hamiltonian systems in two degrees of freedom

Nested attractors near nonlinear centers

Publisher Summary This chapter presents the global results by local averaging for nearly Hamiltonian systems. It presents a space in which the points represent differential equations; a subset consists of Hamiltonian systems, and a further subset of integrable Hamiltonian systems. Integrable systems do not form an open set, either among Hamiltonian systems or in general, and it is natural to ask what sort of systems lay near an integrable system. For the Hamiltonian case, this leads into the theories of Poincare–Birkhoff and Kolmogorov–Arnol’d–Moser. The chapter presents a case of non-Hamiltonian perturbations of an integrable Hamiltonian system. According to a general theorem of Arnol’d, the phase space of an integrable system is divided up by separatrices into regions admitting action/angle variables r, θ in which the system takes the form r = 0, θ = Ω(r). Each torus r = const is invariant and foliated with minimal invariant tori of lower dimension depending on the commensurability relations satisfied by Ω(r).

James Murdock

Papers

Resonance capture in certain nearly Hamiltonian systems

Nearly hamiltonian systems in two degrees of freedom

Nested attractors near nonlinear centers

Global Results by Local Averaging for Nearly Hamiltonian Systems