The Existence of Shilnikov Homoclinic Orbits in the Michelson System: A Computer Assisted Proof
نویسنده
چکیده
In this paper we present a new topological tool which allows to prove the existence of Shilnikov homoclinic or heteroclinic solutions. We present an application of this method to the Michelson system y′′′ + y′ + 0.5y = c [16]. We prove that there exists a countable set of parameter values c for which a pair of the Shilnikov homoclinic orbits to the equilibrium points (±c√2, 0, 0) appear. This result was conjectured by Michelson [16]. We also show that there exists a countable set of parameter values for which there exists a heteroclinic orbit connecting the equilibrium (−c√2, 0, 0) possessing one dimensional unstable manifold with the equilibrium (c √ 2, 0, 0) possessing one dimensional stable manifold. The method used in the proof can be applied to other reversible systems. To verify assumptions of the main topological theorem for the Michelson system we use rigorous computations based on interval arithmetic.
منابع مشابه
Detecting the Shilnikov scenario in a Hopf-Hopf bifurcation with 1:3 resonance
We investigate the behaviour of the primary solutions at a Hopf-Hopf interaction close to a 1:3 resonance. It turns out, that the secondary bifurcations from the primary periodic solution branches are governed by Duffing and Mathieu equations. By numerical path following a homoclinic orbit at a saddle node was detected, giving rise to the Shilnikov scenario. In order to understand the creation ...
متن کاملExistence of Infinitely Many Elliptic Periodic Orbits in Four-Dimensional Symplectic Maps with a Homoclinic Tangency
We study the problem of coexistence of a countable number of periodic orbits of different topological types (saddles, saddle–centers, and elliptic) in the case of four-dimensional symplectic diffeomorphisms with a homoclinic trajectory to a saddle–focus fixed point.
متن کاملHomoclinic Orbits of the FitzHugh-Nagumo Equation: Bifurcations in the Full System
This paper investigates travelling wave solutions of the FitzHugh-Nagumo equation from the viewpoint of fast-slow dynamical systems. These solutions are homoclinic orbits of a three dimensional vector field depending upon system parameters of the FitzHugh-Nagumo model and the wave speed. Champneys et al. [A.R. Champneys, V. Kirk, E. Knobloch, B.E. Oldeman, and J. Sneyd, When Shilnikov meets Hop...
متن کاملHomoclinic chaos in coupled SQUIDs
An rf superconducting quantum interference device (SQUID) consists of a superconducting ring interrupted by a Josephson junction (JJ). The induced supercurrents around the ring are determined by the JJ through the celebrated Josephson relations. We study the dynamics of a pair of parametrically-driven coupled SQUIDs lying on the same plane with their axes in parallel. The drive is through the a...
متن کاملComputer Assisted Proof of the Existence of Homoclinic Tangency for the Hénon Map and for the Forced Damped Pendulum
We present a topological method for the efficient computer assisted verification of the existence of the homoclinic tangency which unfolds generically in a one-parameter family of planar maps. The method has been applied to the Hénon map and the forced damped pendulum ODE.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Foundations of Computational Mathematics
دوره 6 شماره
صفحات -
تاریخ انتشار 2006