Melnikov’s Method Applied to the Double Pendulum
نویسنده
چکیده
Melnikov’s method is applied to the planar double pendulum proving it to be a chaotic system. The parameter space of the double pendulum is discussed, and the integrable cases are identified. In the neighborhood of the integrable case of two uncoupled pendulums Melnikov’s integral is evaluated using residue calculus. In the two limiting cases of one pendulum becoming a rotator or an oscillator, the parameter dependence of chaos, i. e. the width of the separatrix layer is analytically discussed. The results are compared with numerically computed Poincaré surfaces of section, and good agreement is found.
منابع مشابه
Dynamics and Regulation of Locomotion of a Human Swing Leg as a Double-Pendulum Considering Self-Impact Joint Constraint
Background:Despite some successful dynamic simulation of self-impact double pendulum (SIDP)-as humanoid robots legs or arms- studies, there is limited information available about the control of one leg locomotion.Objective :The main goal of this research is to improve the reliability of the mammalians leg locomotion and building more elaborated models close to the natural movements, by modelin...
متن کاملNORMAL FORM SOLUTION OF REDUCED ORDER OSCILLATING SYSTEMS
This paper describes a preliminary investigation into the use of normal form theory for modelling large non-linear dynamical systems. Limit cycle oscillations are determined for simple two-degree-of-freedom double pendulum systems. The double pendulum system is reduced into its centre manifold before computing normal forms. Normal forms are obtained using a period averaging method which is appl...
متن کاملDesign of H-infinity Controller for A Linear Spring Connected Double Inverted Pendulum
A modified double inverted pendulum – modified by connecting the mass carrying the pendulum with another mass through a spring makes the general inverted pendulum become a more interesting problem. The system is defined as a linear spring connected double inverted pendulum as proposed by Hou et al. [1],[2]. The system is highly nonlinear and unstable. However, the system can be simplified to a ...
متن کاملTransformation of Output Constraints in Optimal Control Applied to a Double Pendulum on a Cart
This paper describes a constraint transformation technique for optimal control problems (OCP) with nonlinear single-input single-output (SISO) systems subject to output constraints. An input-output transformation and saturation functions are used to transform the system dynamics into a new unconstrained representation. This method allows to reformulate the original OCP into an unconstrained cou...
متن کاملExponentially small oscillation of 2-dimensional stable and unstable manifolds in 4-dimensional symplectic mappings
Homoclinic bifurcation of 4-dimensional symplectic mappings is asymptotically studied. We construct the 2-dimensional stable and unstable manifolds near the submanifolds which experience exponentially small splitting, and successfully obtain exponentially small oscillating terms in the 2-dimensional manifolds. ∗ E-mail address: [email protected] 1 typeset using PTPTEX.sty <ver...
متن کامل