Spikes Adding to Infinity on Period-1 Orbits to Chaos in the Rössler System

Abstract

In this paper, spikes adding to infinity on bifurcation trees of period-1 orbits to chaos in the Rössler system is studied. The spikes adding on the periodic orbits is completed through a saddle-node bifurcation. With onset of a period-1 orbit, there is 1-spike on such a period-1 orbit, followed by the development from 1-spike to [Formula: see text]-spikes and the period-1 to period-[Formula: see text] orbits have 1-spike to [Formula: see text]-spikes. For a spike bifurcation of a period-1 orbit with [Formula: see text]-spikes ([Formula: see text]), a new spike is added on such a period-1 orbit. Thus, the period-1 orbit has [Formula: see text]-spikes. Such a period-1 to period-[Formula: see text] orbits ([Formula: see text]) have [Formula: see text]-spikes to [Formula: see text]-spikes. The three bifurcation trees of period-1 orbits with [Formula: see text]-spikes ([Formula: see text]) to period-4 orbits with [Formula: see text]-spikes are presented numerically. The phase trajectories and responses of [Formula: see text]-component for period-1 to period-4 orbits with different spikes are given for illustrations of spikes adding on periodic orbits. The spikes adding generating the complexity of period-1 orbits to chaos can be developed.