Fundamental cupolets of chaotic systems

Abstract

Cupolets are a relatively new class of waveforms that represent highly accurate approximations to the unstable periodic orbits of chaotic systems, and large numbers can be efficiently generated via a control method where small kicks are applied along intersections with a control plane. Cupolets exhibit the interesting property that a given set of controls, periodically repeated, will drive the associated chaotic system onto a uniquely defined cupolet regardless of the system’s initial state. We have previously demonstrated a method for efficiently steering from one cupolet to another using a graph-theoretic analysis of the connections between these orbits. In this paper, we discuss how connections between cupolets can be analyzed to show that complicated cupolets are often composed of combinations of simpler cupolets. Hence, it is possible to distinguish cupolets according to their reducibility: a cupolet is classified either as composite, if its orbit can be decomposed into the orbits of other cupolets or as fundamental, if no such decomposition is possible. In doing so, we demonstrate an algorithm that not only classifies each member of a large collection of cupolets as fundamental or composite, but that also determines a minimal set of fundamental cupolets that can exactly reconstruct the orbit of a given composite cupolet. Furthermore, this work introduces a new way to generate higher-order cupolets simply by adjoining fundamental cupolets via sequences of controlled transitions. This allows for large collections of cupolets to be collapsed onto subsets of fundamental cupolets without losing any dynamical information. We conclude by discussing potential future applications.

Cupolets: History, Theory, and Applications

“…During the course of the targeting work, it became apparent that a transition sequence could be made to close up on itself and produce a closed loop around the attractor [71]. By interlacing the control sequences of the cupolets used in the transition sequence, the new orbit was seen to be a cupolet that had been created separately from the control method and was instead constructed by amalgamating the orbits of existing simpler cupolets.…”

“…These observations inspired the development of a new irreducibility property that distinguishes the cupolets whose orbits can be decomposed by sequences of switchable cupolets from those cupolets whose orbits cannot be broken down in this manner [71]. Cupolets for which no decomposing set of cupolets can be found are said to be irreducible with respect to a working library of cupolets and are called fundamental cupolets.…”

“…These three fundamental cupolets are pairwise switchable, and so their orbits can be spliced together at their intersections with transient-free controlled transitions that collectively assemble C0001111101110111. By doing so, the fundamental cupolets give the fundamental decomposition for this cupolet (see [71] for more details). The symbols shown along the cupolets' orbits of Figure 12 serve simply to further distinguish the orbits of these cupolets.…”

“…The fundamental cupolets effectively serve as basis-like building blocks for higher-order cupolets, which allows for a significant portion of a cupolet library to be completely collapsed onto its set of fundamental cupolets without incurring any information loss. For instance, the study described in [71] utilized a library of 13,896 cupolets, of which 2414 were composite cupolets whose fundamental decompositions were completely determined by the algorithm. This means that 2414/13,896 ≈ 17% of the original cupolet library was superfluous and could be discarded without losing any dynamical information.…”

Chaotic Entanglement: Entropy and Geometry

“…This allowed methods of graph theory to be brought to bear to develop an algorithm to calculate an efficient, in some sense, low-energy transition from an initial trajectory to a target trajectory by riding along and transitioning between cupolets [12]. Further investigation revealed that certain cupolets could be defined as fundamental and others as composite, with the fundamental cupolets irreducible within a finite space of cupolets [13].…”