882 research outputs found
Asymptotic stability in singular perturbation problems. II: Problems having matched asymptotic expansion solutions
AbstractThe stability of systems of ordinary differential equations of the form dxdt = f(t, x, y, ϵ), ϵ dydt = g(t, x, y, ϵ), where ϵ is a real parameter near zero, is studied. It is shown that if the reduced problem dxdt = f(t, x, y, 0), 0 = g(t, x, y, 0), is stable, and certain other conditions which ensure that the method of matched asymptotic expansions can be used to construct solutions are satisfied, then the full problem is asymptotically stable as t → ∞, and a domain of stability is determined which is independent of ϵ. Moreover, under certain additional conditions, it is shown that the solutions of the perturbed problem have limits as t → ∞. In this case, it is shown how these limits can be calculated directly from the equations f(∞, x, y, ϵ) = 0 g(∞, x, y, ϵ) = 0 as expansions in powers of ϵ
Slowly modulated oscillations in nonlinear diffusion processes
It is shown here that certain systems of nonlinear (parabolic) reaction-diffusion equations have solutions which are approximated by oscillatory functions in the form R(ξ - cτ)P(t^*) where P(t^*) represents a sinusoidal oscillation on a fast time scale t* and R(ξ - cτ) represents a slowly-varying modulating amplitude on slow space (ξ) and slow time (τ) scales. Such solutions describe phenomena in chemical reactors, chemical and biological reactions, and in other media where a stable oscillation at each point (or site) undergoes a slow amplitude change due to diffusion
Plasticity and learning in a network of coupled phase oscillators
A generalized Kuramoto model of coupled phase oscillators with slowly varying
coupling matrix is studied. The dynamics of the coupling coefficients is driven
by the phase difference of pairs of oscillators in such a way that the coupling
strengthens for synchronized oscillators and weakens for non-synchronized
pairs. The system possesses a family of stable solutions corresponding to
synchronized clusters of different sizes. A particular cluster can be formed by
applying external driving at a given frequency to a group of oscillators. Once
established, the synchronized state is robust against noise and small
variations in natural frequencies. The phase differences between oscillators
within the synchronized cluster can be used for information storage and
retrieval.Comment: 10 page
Synchronization of oscillators coupled through an environment
We study synchronization of oscillators that are indirectly coupled through
their interaction with an environment. We give criteria for the stability or
instability of a synchronized oscillation. Using these criteria we investigate
synchronization of systems of oscillators which are weakly coupled, in the
sense that the influence of the oscillators on the environment is weak. We
prove that arbitrarily weak coupling will synchronize the oscillators, provided
that this coupling is of the 'right' sign. We illustrate our general results by
applications to a model of coupled GnRH neuron oscillators proposed by Khadra
and Li, and to indirectly weakly-coupled lambda-omega oscillator
Multistable attractors in a network of phase oscillators with three-body interaction
Three-body interactions have been found in physics, biology, and sociology.
To investigate their effect on dynamical systems, as a first step, we study
numerically and theoretically a system of phase oscillators with three-body
interaction. As a result, an infinite number of multistable synchronized states
appear above a critical coupling strength, while a stable incoherent state
always exists for any coupling strength. Owing to the infinite multistability,
the degree of synchrony in asymptotic state can vary continuously within some
range depending on the initial phase pattern.Comment: 5 pages, 3 figure
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