ISSN:
1573-269X
Keywords:
Chaos
;
perturbation methods
;
elliptic functions
;
differential equations
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract We investigate the system $$\ddot x - x\cos \varepsilon 1 + x^3 = 0$$ in which ε≪1 by using averaging and elliptie functions. It is shown that this system is applicable to the dynamics of the familiar rotating-plane pendulum. The slow foreing permits us to envision an ‘instantancous phase portrait’ in the $$x - \dot x$$ phase plane which exhibits a center at the origin when cos ε1≤0 and a saddle and associated double homoclinic loop separatrix when cos ɛ 1 〉 0. The chaos in this problem is related to the question of on which side (left (=L) or right (=R)) of the reappearing double homoclinic loop separatrix a motion finds itself. We show that the sequence of L's and R's exhibits sensitive dependence on initial conditions by using a simplified model which assumes that motions cross the instantancous separatrix instantancously. We also present an improved model which ‘patches’ a separatrix boundary layer onto the averaging model. The predictions of both models are compared with the results of numerical integration.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01893171
