scholarly journals Chaos in an anharmonic oscillator

1995 ◽  
Vol 37 (2) ◽  
pp. 186-207 ◽  
Author(s):  
J. R. Christie ◽  
K. Gopalsamy
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Autonomous System ◽  
Anharmonic Oscillator ◽  
Homoclinic Orbits ◽  
Chaotic Behaviour ◽  
Differential Systems ◽  
Melnikov's Method ◽  
One Dimensional ◽  
Melnikov’S Method

AbstractUsing Melnikov's method, the existence of chaotic behaviour in the sense of Smale in a particular time-periodically perturbed planar autonomous system of ordinary differential equations is established. Examples of planar autonomous differential systems with homoclinic orbits are provided, and an application to the dynamics of a one-dimensional anharmonic oscillator is given.

scholarly journals Subharmonic orbits in an anharmonic oscillator

1997 ◽  
Vol 38 (3) ◽  
pp. 307-315 ◽  
Author(s):  
J. R. Christie ◽  
K. Gopalsamy ◽  
M. P. Panizza
Keyword(s):  
Differential Equations ◽  
Autonomous System ◽  
Anharmonic Oscillator ◽  
Dynamical Behaviour ◽  
Sufficient Condition ◽  
One Dimensional ◽  
Perturbed System ◽  
Melnikov Theory ◽  
The One ◽  
Subharmonic Orbits

AbstractIn a recent paper, Christie and Gopalsamy [2] used Melnikov's method to establish a sufficient condition for the existence of chaotic behaviour, in the sense of Smale, in a particular time-periodically perturbed planar autonomous system of ordinary differential equations. They then concluded with an application to the dynamics of a one-dimensional anharmonic oscillator. In this paper, the same system is considered and a condition for the existence of subharmonic orbits in the perturbed system is deduced, using the subharmonic Melnikov theory. Finally, an application is given to the dynamical behaviour of the one-dimensional anharmonic oscillator system.

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

The Poincaré-Cartan forms of one-dimensional variational integrals

2020 ◽  
Vol 70 (6) ◽  
pp. 1381-1412
Author(s):  
Veronika Chrastinová ◽  
Václav Tryhuk
Keyword(s):  
Differential Equations ◽  
Variational Problem ◽  
Ordinary Differential Equations ◽  
Differential Forms ◽  
Point Of View ◽  
Full Generality ◽  
One Dimensional ◽  
Variational Integrals

AbstractFundamental concepts for variational integrals evaluated on the solutions of a system of ordinary differential equations are revised. The variations, stationarity, extremals and especially the Poincaré-Cartan differential forms are relieved of all additional structures and subject to the equivalences and symmetries in the widest possible sense. Theory of the classical Lagrange variational problem eventually appears in full generality. It is presented from the differential forms point of view and does not require any intricate geometry.

scholarly journals Linearly Coupled Anharmonic Oscillators and Integrability

1987 ◽  
Vol 40 (5) ◽  
pp. 587
Author(s):  
W-H Steeb ◽  
JA Louw ◽  
CM Villet
Keyword(s):  
Lyapunov Exponent ◽  
Anharmonic Oscillator ◽  
First Integrals ◽  
Chaotic Behaviour ◽  
Painlevé Test ◽  
Anharmonic Oscillators ◽  
One Dimensional

The Painleve test for a linearly coupled anharmonic oscillator is performed. We show that this system does not pass the Painleve test. This suggests that this system is not integrable. Moreover, we apply Ziglin's (1983) theorem which provides a criterion for non-existence of first integrals besides the Hamiltonian. Calculating numerically the maximal one-dimensional Lyapunov exponent, we find regions with positive exponents. Thus, the system can show chaotic behaviour. Finally we compare our results with the quartic coupled anharmonic oscillator.

Chaos in a model of forced quasi-geostrophic flow over topography: an application of Melnikov's method

1991 ◽  
Vol 226 ◽  
pp. 511-547 ◽  
Author(s):  
J. S. Allen ◽  
R. M. Samelson ◽  
P. A. Newberger
Keyword(s):  
Periodic Orbit ◽  
Saddle Point ◽  
Wind Stress ◽  
Homoclinic Orbits ◽  
Periodic Forcing ◽  
Melnikov's Method ◽  
Geostrophic Flow ◽  
Melnikov’S Method ◽  
Parameter Values ◽  
Time Periodic

We demonstrate the existence of a chaotic invariant set of solutions of an idealized model for wind-forced quasi-geostrophic flow over a continental margin with variable topography. The model (originally formulated to investigate mean flow generation by topographic wave drag) has bottom topography that slopes linearly offshore and varies sinusoidally alongshore. The alongshore topographic scales are taken to be short compared to the cross-shelf scale, allowing Hart's (1979) quasi-two-dimensional approximation, and the governing equations reduce to a non-autonomous system of three coupled nonlinear ordinary differential equations. For weak (constant plus time-periodic) forcing and weak friction, we apply a recent extension (Wiggins & Holmes 1987) of the method of Melnikov (1963) to test for the existence of transverse homoclinic orbits in the model. The inviscid unforced equations have two constants of motion, corresponding to energy E and enstrophy M, and reduce to a one-degree-of-freedom Hamiltonian system which, for a range of values of the constant G = E − M, has a pair of homoclinic orbits to a hyperbolic saddle point. Weak forcing and friction cause slow variations in G, but for a range of parameter values one saddle point is shown to persist as a hyperbolic periodic orbit and Melnikov's method may be applied to study the perturbations of the associated homoclinic orbits. In the absence of time-periodic forcing, the hyperbolic periodic orbit reduces to the unstable fixed point that occurs with steady forcing and friction. The method yields analytical expressions for the parameter values for which sets of chaotic solutions exist for sufficiently weak time-dependent forcing and friction. The predictions of the perturbation analysis are verified numerically with computations of Poincaré sections for solutions in the stable and unstable manifolds of the hyperbolic periodic orbit and with computations of solutions for general initial-value problems. In the presence of constant positive wind stress τ0 (equatorward on eastern ocean boundaries), chaotic solutions exist when the ratio of the oscillatory wind stress τ1 to the bottom friction parameter r is above a critical value that depends on τ0/r and the bottom topographic height. The analysis complements a previous study of this model (Samelson & Allen 1987), in which chaotic solutions were observed numerically for weak near-resonant forcing and weak friction.

Smoothness of Asymptotic Phase Revisited

2011 ◽  
Vol 11 (4) ◽  
Author(s):  
Flaviano Battelli ◽  
Kenneth J. Palmer
Keyword(s):  
Periodic Solution ◽  
Vector Field ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stable Manifold ◽  
Autonomous System ◽  
Asymptotic Phase

AbstractIt is well-known that solutions on the stable manifold of a hyperbolic periodic solution of an autonomous system of ordinary differential equations have an asymptotic phase which has the same order of smoothness as the vector field. In this paper we show if the system depends on a parameter that, in general, the asymptotic phase loses one order of smoothness in the parameter.

scholarly journals Quotients of Euler Equations on Space Curves

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 186
Author(s):  
Anna Duyunova ◽  
Valentin Lychagin ◽  
Sergey Tychkov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Euler Equations ◽  
Gas Flow ◽  
Euler System ◽  
Partial Differential ◽  
One Dimensional ◽  
Space Curves ◽  
Ode System

Quotients of partial differential equations are discussed. The quotient equation for the Euler system describing a one-dimensional gas flow on a space curve is found. An example of using the quotient to solve the Euler system is given. Using virial expansion of the Planck potential, we reduce the quotient equation to a series of systems of ordinary differential equations (ODEs). Possible solutions of the ODE system are discussed.

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