Energy flow-guided synchronization between chaotic circuits

Abstract

A variety of nonlinear electric devices are suitable for building chaotic circuits by setting appropriate values for parameters in the linear and nonlinear electronic components. The capacitor and induction coil can contain time-varying physical field energy when the time-varying current passed across these devices. When the outputs end of two chaotic circuits are connected via certain coupling devices, energy flow can be propagated bidirectionally and complete synchronization is reached when the energy pumping from the outputs end is balanced. In this paper, a nonreversing diode is connected into the coupling channel to adjust the energy flow via the coupling components (resistor, capacitor, induction coil), which connects the outputs end of two Chua circuits (in chaotic or periodical states). According to Helmholtz’ theorem, the Hamilton energy of the dimensionless chaotic systems under unidirectional coupling is calculated. When the two coupled Chua systems are selected with different initial values, they contain different initial field energies because the Hamilton energy is dependent on the variables of this system completely. When energy flow is propagated and pumped along the coupling channel, synchronization can be realized. Otherwise, synchronization approach is blocked even the coupling intensity is increased greatly. In this work, a single coupling channel connecting a nonreversing diode and resistor (capacitor, induction coil, respectively) in series is built to explore synchronization approach. The results show that the synchronization realization is dependent on the coupling devices and circuit verification is also supplied. It gives important clues to understand the synchronization mechanism in a forward neural network.

Introduction

Nonlinear oscillators can present a variety of oscillation modes by adjusting the intrinsic parameters and/or imposing appropriate external stimulus. On the other hand, the observable variables in the complex systems can be detected and estimated to build reliable dynamical models, which can be further mapped into nonlinear oscillators after applying appropriate scale transformation on these physical variables and parameters. For isolated nonlinear oscillator, nonlinear analysis such as bifurcation calculation provides effective way to capture important information and predict possible mode transition in the sampled series. For more than two nonlinear oscillators and circuits, analysis of stability in synchronization is helpful to know the information encoding and exchange, and pumping in energy flow. Chaotic circuits are often used as feasible signal generator for producing a variety of periodic signals and chaotic series for selecting secure keys in communications and even image encryption [1], [2], [3], [4]. Indeed, the occurrence of chaos in biological systems indicates the self-adaption by triggering different modes in oscillation of the observable variables. For example, neurons can present quiescent, spiking, bursting and even chaotic firing patterns under different stimuli. In this way, neurons and neural network can select the most suitable firing mode to give rapid response to adjacent or external stimulus [5], [6], [7]. Most of the chaotic systems can be controlled and coupled to reach desired orbits and then the output series can be estimated by applying many effective schemes [8], [9], [10]. In fact, the investigation of chaotic circuits has another important application in computational neuroscience than general synchronization for secure communication and image encryption. That is, neuron can be regarded as intelligent signal processor which can capture signals from many channels, and nonlinear circuits [11], [12], [13], [14], [15] can be tamed as artificial neural circuits for signal encoding. Furthermore, collective behaviors in neural networks composed of neural circuits [16], [17], [18] can be estimated to predict the occurrence of nervous disease.

Bidirectional coupling between chaotic circuits shows its reliability for reaching synchronization because time-varying current propagated along the coupling channel can be thought as adaptive stimulus or driving, and then both of the chaotic circuits are driven to keep pace with the external forcing. When complete synchronization is realized, the current along the coupling channel is terminated because the voltage from the outputs end is balanced completely. For nonlinear oscillators and networks, direct variable coupling is often applied to find emergence of synchronization and pattern selection [19], [20], [21], [22], [23], and it accounts for electrical synapse coupling between neurons [24], [25], [26], [27]. From physical viewpoint, this kind of variable coupling results from direct voltage coupling via linear resistor connection between nonlinear circuits. In fact, most of the electronic components such as resistor, memristor [28], [29], [30], [31], capacitor and induction coil can be used to couple nonlinear circuits by bridging different coupling channels. In particular, capacitor connection between chaotic circuits explains the physical mechanism for differential coupling control [32], [33], [34] and time-varying electric field is generated in the coupling capacitor by pumping energy from the coupled circuits. On the other hand, magnetic field coupling via induction coil explains the physical mechanism for integral coupling control [35], [36], [37] and time-varying magnetic field is induced in the coupling coil by pumping energy from the coupled circuits. In a word, both capacitor coupling and induction coil coupling just activate a kind of field coupling [38] between chaotic circuits or neural circuits because time-varying physical field is induced in the coupling devices. In addition, when induction current passed along the coupling devices, energy consumption and release occur. For example, the coupling resistor consumes Joule heat to balance the outputs energy from the coupled circuits. The coupling capacitor can collect and keep electric field energy while the coupling induction coil just captures magnetic field energy before reaching complete synchronization. Therefore, the estimation of energy flow along the coupling devices gives important clues to understand the coupling synchronization [39] from physical viewpoint.

In this paper, the known Chua circuit [40,41] is used for synchronization investigation. Two chaotic Chua circuits are connected with one coupling channel by inserting a nonreversing diode, which can control the direction of energy flow and current. Hamilton energy [42], [43], [44], [45], [46] for the dimensionless Chua system, coupling devices is estimated, respectively. The coupling channel is built by using resistor, capacitor, and induction coil, respectively. Furthermore, a nonreversing diode is connected with these coupling devices in series thus the energy flow can be controlled by the ideal diode with zero turn-on voltage. The two coupled Chua systems are selected with different initial values and coupling channel is activated to evaluate reliability of synchronization realization.