Linearities of the f( ) Spectrum at Bifurcations of Chaos in Dissipative Differential Systems

Abstract

It is shown that the singularity spectrum I(a) of the natural invariant measure in the driven damped pendulum exhibits three types of linear slopes at bifurcation points of chaos, which agree with theoretical estimates of the linear slopes . . Chaotic attractors of dissipative differential systems have fascinating multi­ fractal structures. I) Recently it has been shown that the structures could be char­ acterized by the generalized dimensions D(q), (-=<q<=) and the singularity spectra I(a) of the natural invariant measures. 2 ) For two-dimensional maps, it has been found that D(q) and I(a) have close relation to the q-weighted average A(q) of the coarse-grained expansion rates A of nearby orbits along the local unstable manifolds and their spectrum ¢I(A).3H) In our previous papers,6H) therefore, chaotic attractors at ~arious bifurcation points have been studied in terms of A(q) and ¢I(A), and singular local structures of chaotic attractors have been shown to bring about linearities in ¢I(A) which cause q-phase transitions of A(q). For two-dimensional maps, it has also been shown that the linearities in ¢I(A) lead to linearities in I(a) which causeq-phase transitions of D(q).IO) This indicates that the metric structures of chaotic attractors can be characterized by the linearities in l(a).5) In this paper, therefore, we shall investigate the I(a) spectrum and its linearities for dissipative differential systems. As a typical example of dissipative differential systems which exhibit chaotic behavior, we take the driven damped pendulum, whose equation of motion is