A Partial Differential Equation with Infinitely many Periodic Orbits: Chaotic Oscillations of a Forced Beam
نویسندگان
چکیده
This paper delineates a class of time-periodically perturved evolution equations in a Banach space whose associated Poincaré map contains a Smale horseshoe. This implies that such systems possess periodic orbits with arbitrarily high period. The method uses techniques originally due to Melnikov and applies to systems of the form ẋ = f0(x) + εf1(x, t), where ẋ = f0(x) is Hamiltonian and has a homoclinic orbit. We give an example from structural mechanics: sinusoidally forced vibrations of a buckled beam.
منابع مشابه
Forced oscillations of a damped Korteweg-de Vries equation on a periodic domain
In this paper, we investigate a damped Korteweg-de Vries equation with forcing on a periodic domain $mathbb{T}=mathbb{R}/(2pimathbb{Z})$. We can obtain that if the forcing is periodic with small amplitude, then the solution becomes eventually time-periodic.
متن کاملConnecting Period-Doubling Cascades to Chaos
The appearance of infinitely-many period-doubling cascades is one of the most prominent features observed in the study of maps depending on a parameter. They are associated with chaotic behavior, since bifurcation diagrams of a map with a parameter often reveal a complicated intermingling of period-doubling cascades and chaos. Period doubling can be studied at three levels of complexity. The fi...
متن کاملEvolutionary Synthesis Of Control Law For Higher Periodic Orbits Of Chaotic Logistic Equation
This research deals with a synthesis of control law for selected discrete chaotic system by means of analytic programming. The novelty of the approach is that a tool for symbolic regression – analytic programming is used for the purpose of stabilization of higher periodic orbits – oscillations between several values of chaotic system. The paper consists of the descriptions of analytic programmi...
متن کاملInvestigation and Control of Unstable Chaotic Behavior Using of Chaos Theory in Two Electrical Power Systems: 1-Buck Converter2- Power Transformer
This paper consist of two sections: control and stabilizing approach for chaotic behaviour of converter is introduced in first section of this paper for the removal of harmonic caused by the chaotic behaviour in current converter. For this work, a Time- Delayed Feedback Controller (TDFC) control method for stability chaotic behaviour of buck converter for switching courses in current control mo...
متن کاملChaotic Behavior in Differential Equations Driven by a Brownian Motion
In this paper, we investigate the chaotic behavior of ordinary differential equations with a homoclinic orbit to a dissipative saddle point under an unbounded random forcing driven by a Brownian motion. We prove that, for almost all sample pathes of the Brownian motion in the classical Wiener space, the forced equation admits a topological horseshoe of infinitely many branches. This result is t...
متن کامل