Signatures of Quantum Mechanics in Chaotic Systems

Abstract

We consider the quantum-classical correspondence from a classical perspective by discussing the potential for chaotic systems to support behaviors normally associated with quantum mechanical systems. Our main analytical tool is a chaotic system's set of cupolets, which are essentially highly-accurate stabilizations of its unstable periodic orbits. The discussion is motivated by the bound or entangled states that we have recently detected between interacting chaotic systems, wherein pairs of cupolets are induced into a state of mutually-sustaining stabilization that can be maintained without external intervention. This state is known as chaotic entanglement as it has been shown to exhibit several properties consistent with quantum entanglement. For instance, should the interaction be disturbed, then the chaotic entanglement would be broken. In this paper, we further describe chaotic entanglement and go on to discuss the capacity for chaotic systems to exhibit other characteristics that are conventionally associated with quantum mechanics, namely analogs to wave function collapse, the measurement problem, the superposition of states, and to quantum entropy definitions. In doing so, we argue that these characteristics need not be regarded exclusively as quantum mechanical.

Chaotic Entanglement: Entropy and Geometry

“…The chaotic entanglement summarized in this section has been found with numerous forms of exchange functions and many different cupolet pairs, as shown in the references [14,15]. Another example has been found where the exchange function is the identity function, so the applied controls are determined purely by the lobe visitation sequence of the cupolets.…”

“…The entanglement is persistent, but if one cupolet is disturbed from its periodic orbit, it subsequently affects the stability of the partner cupolet, and vice versa. This, and related properties evoke several connections to quantum entanglement, as discussed in [15]. In the cited references, hundreds of entangled cupolet pairs have been identified for the double scroll system, using several classes of exchange functions.…”

“…In a recent series of papers [14,15,21], it is shown that pairs of chaotic systems that are capable of producing cupolets may be able to interact in a way that causes them to chaotically entangle, by which we mean that through their interaction they fall into a state of mutual stabilization onto their cupolets. For this to occur, the two chaotic systems are assumed to be connected via an exchange function that exchanges control information and, in a manner that will be detailed below, when one system is induced to collapse onto a cupolet state, that cupolet state passes into the interaction function which then induces the second chaotic system to collapse onto its own cupolet state.…”

“…To make this visually evident, the following figures are reprinted from [15], with a summarized discussion. Consider the two cupolets shown in Figure 3, along with the cartoon depicted in Figure 4, which portrays the chaotic entanglement of cupolets C000011111 and C000011111 in a series of step-by-step illustrations.…”

“…It was shown in [14] that such mutually stabilizing persistent states do indeed exist, and this type of interaction was termed chaotic entanglement. Further discussions of chaotic entanglement and comparisons with quantum entanglement are covered in [15]. This will be the jumping off point for the work in this paper, so more review is provided in Section 2 below.…”