Damon A. Miller

Theory and experimental realization of observer-based discrete-time hyperchaos synchronization

Research in the synchronization of dynamical systems has been mainly focused on chaotic rather than hyperchaotic systems and on continuous-time rather than discrete-time systems. Numerical simulations dominate these studies and results typically lack experimental data. This brief fills these gaps by 1) presenting a technique for the exact (dead-beat) synchronization of hyperchaotic discrete-time systems; and 2) describing an electronic implementation of this technique for the generalized Henon map. The synchronization strategy is based on the observer concept and enables a wide class of hyperchaotic discrete-time systems to be synchronized via a scalar signal. An electronic implementation provides verification of the theoretical results and confirms the feasibility of realizing this approach in hardware.

Experimental realization of observer-based hyperchaos synchronization

Research literature on the topic of synchronizing hyperchaotic systems predominantly addresses theoretical issues-most investigations lack experimental results. This paper aims to bridge this gap by presenting an electronic circuit implementation of a particular hyperchaotic synchronization strategy. Specifically, previous research has demonstrated that linear observers may be synchronized to a certain class of hyperchaotic systems via a scalar signal. This paper describes an experimental realization of this type of linear observer for the four-dimensional hyperchaotic system proposed by Tamasevicius et al. For comparative purposes, the linear observer circuitry may be electronically reconfigured as a second resistively coupled hyperchaotic oscillator. Experimental results, provided for both cases, verify theoretical results, demonstrate the feasibility of electronically implementing linear observers for hyperchaotic oscillators, and suggest that the linear observer is robust with respect to component mismatches between the hyperchaotic oscillator and the linear observer.

A discrete generalized hyperchaotic Henon map circuit

This paper describes a discrete-component electronic implementation of a discrete-time hyperchaotic generalized Henon map with analog states. The relatively simple circuit design uses commonly available parts and is readily constructed. Initial analysis of experimental results indicates that the circuit is functional.

A dynamical system perspective of structural learning with forgetting

Structural learning with forgetting is an established method of using Laplace regularization to generate skeletal artificial neural networks. In this paper we develop a continuous dynamical system model of regularization in which the associated regularization parameter is generalized to be a time-varying function. Analytic results are obtained for a Laplace regularizer and a quadratic error surface by solving a different linear system in each region of the weight space. This model also enables a comparison of Laplace and Gaussian regularization. Both of these regularizers have a greater effect in weight space directions which are less important for minimization of a quadratic error function. However, for the Gaussian regularizer, the regularization parameter modifies the associated linear system eigenvalues, in contrast to its function as a control input in the Laplace case. This difference provides additional evidence for the superiority of the Laplace over the Gaussian regularizer.

Generation of a four-wing chaotic attractor by two weakly-coupled lorenz systems

This paper presents a novel chaotic four-wing attractor generated by coupling two identical Lorenz systems. An analysis of the proposed system shows that its equilibria have certain symmetries with respect to specific coordinate planes and the eigenvalues of the associated Jacobian matrices exhibit the property of similarity. In analogy with the original Lorenz system, where the two wings of the butterfly attractor are located around the two equilibria with the unstable pair of complex-conjugate eigenvalues, this paper shows that the four wings of this new attractor are located around four equilibria with four unstable complex-conjugate eigenvalues. A generalization of the proposed system to realize an eight-wing attractor is also described.

PROJECTIVE SYNCHRONIZATION VIA A LINEAR OBSERVER: APPLICATION TO TIME-DELAY, CONTINUOUS-TIME AND DISCRETE-TIME SYSTEMS

This Letter presents a general approach to projective synchronization that features a linear observer with an ability to arbitrarily scale a drive system attractor. The technique can be applied to wide classes of chaotic and hyperchaotic systems, namely time-delay systems described by functional differential equations (FDEs), continuous-time systems described by ordinary differential equations (ODEs) and discrete-time systems described by difference equations (DEs). Theoretical and simulation results demonstrate that a linear observer can duplicate chaotic system states in any desired scale using only a scalar synchronizing signal. The proposed approach is readily implemented in hardware.

Synchronization and anti-synchronization of Chua's oscillators via a piecewise linear coupling circuit

A chaotic associative memory may be constructed by coupling a network of Chuas circuits via piecewise linear conductances. Synchronization and anti-synchronization states are used to represent binary memory patterns. The chaotic network dynamics enable the memory to wander among patterns which have non-zero correlations with the input pattern. This paper describes two discrete circuits which may be used as the basis for implementing the coupling element as proposed in Jankowski et al. (1995). Initial coupling experiments support the proposed design approaches,.

Pruning via Dynamic Adaptation of the Forgetting Rate in Structural Learning

Structural learning adds a constant weight decay term to the standard backpropagation update rule. At the conclusion of training an initially oversized multilayer feedforward neural network has typically been reduced to near the minimum size required to accomplish a desired mapping and thus this method offers effective pruning. The degree of weight decay is determined by a forgetting rate E which is critical to both learning and pruning success. The choice of E requires careful consideration and current selection criteria are either inexact or computationally expensive. This paper considers E to be a dynamic parameter which can be used to control learning in order to provide redundant weights sufficient time to decay to near zero values. This approach yields pruning results comparable or superior to those obtained by considering E to be constant while reducing computational expense and eliminating the need to determine an optimal value of E .

Synchronizing chaotic systems up to an arbitrary scaling matrix via a single signal

Abstract This paper introduces a novel type of synchronization, where two chaotic systems synchronize up to an arbitrary scaling matrix. In particular, each drive system state synchronizes with a linear combination of response system states by using a single synchronizing signal. The proposed observer-based method exploits a theorem that assures asymptotic synchronization for a wide class of continuous-time chaotic (hyperchaotic) systems. Two examples, involving Rossler’s system and a hyperchaotic oscillator, show that the proposed technique is a general framework to achieve any type of synchronization defined to date.

Exploring optimal current stimuli that provide membrane voltage tracking in a neuron model

Studying neurons from an energy efficiency perspective has produced results in the research literature. This paper presents a method that enables computation of low energy input current stimuli that are able to drive a reduced Hodgkin–Huxley neuron model to approximate a prescribed time-varying reference membrane voltage. An optimal control technique is used to discover an input current that optimally minimizes a user selected balance between the square of the input stimulus current (input current ‘energy’) and the difference between the reference voltage and the membrane voltage (tracking error) over a stimulation period. Selecting reference signals to be membrane voltages produced by the neuron model in response to common types of input currents i(t) enables a comparison between i(t) and the determined optimal current stimulus i*(t). The intent is not to modify neuron dynamics, but through comparison of i(t) and i*(t) provide insight into neuron dynamics. Simulation results for four different bifurcation types demonstrate that this method consistently finds lower energy stimulus currents i*(t) that are able to approximate membrane voltages as produced by higher energy input currents i(t) in this neuron model.