Controlled transitions between cupolets of chaotic systems

Abstract

We present an efficient control scheme that stabilizes the unstable periodic orbits of a chaotic system. The resulting orbits are known as cupolets and collectively provide an important skeleton for the dynamical system. Cupolets exhibit the interesting property that a given sequence of controls will uniquely identify a cupolet, regardless of the system's initial state. This makes it possible to transition between cupolets, and thus unstable periodic orbits, simply by switching control sequences. We demonstrate that although these transitions require minimal controls, they may also involve significant chaotic transients unless carefully controlled. As a result, we present an effective technique that relies on Dijkstra's shortest path algorithm from algebraic graph theory to minimize the transients and also to induce certainty into the control of nonlinear systems, effectively providing an efficient algorithm for the steering and targeting of chaotic systems. Efficiently directing a dynamical system to a desired state is a goal of many engineering applications. This process is known as "targeting" and is often achievable using small nonlinear controls. For a chaotic system, the goal of many control methods is either to stabilize the system onto one of its many unstable periodic orbits (UPOs) or to induce the system to transition between its UPOs until a targeted orbit is attained. The set of UPOs plays a significant role in determining the dynamics of a chaotic system and is said to form the skeleton of an associated attractor. In this paper, we combine a particularly effective chaos control method with algebraic graph theory and present a new technique allowing for the efficient transitioning between periodic orbits of chaotic systems. This allows one to navigate efficiently around a chaotic attractor and in physical applications conserve energy and power.

Signatures of Quantum Mechanics in Chaotic Systems

“…This is easier to visualize in many low-dimensional chaotic systems, such as the double scroll, Lorenz, and Rössler systems, where the attractors are locally ribbon-like in at least part of their domains. Cupolets are generated in such a way that uniqueness considerations also apply, except possibly at certain locations along a control plane where the controls are applied [ 29 ]. Chaotic trajectories are thus restricted to evolving along unique paths that are locally bounded by UPOs and cupolets, which means that the dynamics of chaotic systems are locally dependent on these orbits.…”

“…In this section, we summarize first the control technique that is used to generate cupolets and then the applications of cupolets that have particular relevance to our chaotic entanglement research. More technical details of the control process can be found in [ 22 , 23 , 25 , 28 , 29 ].…”

“…Cupolet restabilization is guaranteed because of the injective relationship that exists between cupolets and the binary control sequences. This makes it possible to transition between cupolets, and thus between UPOs, simply by switching control sequences [ 29 ].…”

On the Potential for Entangled States Between Chaotic Systems

“…This not only minimizes the accumulation of round-off error and the system's sensitivity to initial conditions, but also suggests that some cupolets may intersect. This intersection would occur exactly in the center of a bin and can be used in some applications to jump from one UPO to another [Morena et al, 2014].…”

“…What makes cupolets so practical is that very little information is required in stabilizing their orbits, the diversity of their frequency spectra, and how accessible their dynamical behaviors are via small controls. It has recently been discovered how to transition efficiently between cupolets and hence how to navigate around a chaotic attractor [Morena et al, 2014]. For organizational purposes, we adopt a naming convention for the cupolets that incorporate the control codes used to generate them.…”