Signatures of Quantum Mechanics in Chaotic Systems

Cupolets in a chaotic neuron model

This paper reports the first finding of cupolets in a chaotic Hindmarsh-Rose neural model. Cupolets (chaotic, unstable, periodic, orbit-lets) are unstable periodic orbits that have been stabilized through a particular control scheme by applying a binary control sequence. We demonstrate different neural dynamics (periodic or chaotic) of the Hindmarsh-Rose model through a bifurcation diagram where the external input current, I, is the bifurcation parameter. We select a region in the chaotic parameter space and provide the results of numerical simulations. In this chosen parameter space, a control scheme is applied when the trajectory intersects with either of the two control planes. The type of the control is determined by a bit in a binary control sequence. The control is either a small microcontrol (0) or a large macrocontrol (1) that adjusts the future dynamics of the trajectory by a perturbation determined by the coding function _r ( x ). We report the discovery of many cupolets with corresponding control sequences and comment on the differences with previously reported cupolets in the double scroll system. We provide some examples of the generated cupolets and conclude by discussing potential implications for biological neurons.

Chaotic Entanglement: Entropy and Geometry

In chaotic entanglement, pairs of interacting classically-chaotic systems are induced into a state of mutual stabilization that can be maintained without external controls and that exhibits several properties consistent with quantum entanglement. In such a state, the chaotic behavior of each system is stabilized onto one of the system’s many unstable periodic orbits (generally located densely on the associated attractor), and the ensuing periodicity of each system is sustained by the symbolic dynamics of its partner system, and vice versa. Notably, chaotic entanglement is an entropy-reversing event: the entropy of each member of an entangled pair decreases to zero when each system collapses onto a given period orbit. In this paper, we discuss the role that entropy plays in chaotic entanglement. We also describe the geometry that arises when pairs of entangled chaotic systems organize into coherent structures that range in complexity from simple tripartite lattices to more involved patterns. We conclude with a discussion of future research directions.

Fundamental cupolets of chaotic systems

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.

Sigmoidal synaptic learning produces mutual stabilization in chaotic FitzHugh-Nagumo model

This paper investigates the interaction between two coupled neurons at the terminal end of a long chain of neurons. Specifically, we examine a bidirectional, two-cell FitzHugh-Nagumo neural model capable of exhibiting chaotic dynamics. Analysis of this model shows how mutual stabilization of the chaotic dynamics can occur through sigmoidal synaptic learning. Initially, this paper begins with a bifurcation analysis of an adapted version of a previously studied FitzHugh-Nagumo model that indicates regions of periodic and chaotic behaviors. Through allowing the synaptic properties to change dynamically via neural learning, it is shown how the system can evolve from chaotic to stable periodic behavior. The driving factor between this transition is representative of a stimulus coming down a long neural pathway. The result that two chaotic neurons can mutually stabilize via a synaptic learning implies that this may be a mechanism whereby neurons can transition from a disordered, chaotic state to a stable, ordered periodic state that persists. This approach shows that even at the simplest level of two terminal neurons, chaotic behavior can become stable, sustained periodic behavior. This is achieved without the need for a large network of neurons.