Cecilia Sarasola

FEEDBACK SYNCHRONIZATION OF CHAOTIC SYSTEMS

Feedback coupling through an interaction term proportional to the difference in the value of some behavioral characteristics of two systems is a very common structural setting that leads to synchronization of the behavior of both systems. The degree of synchronization attained depends on the strength of the interaction term and on the mutual interdependency of the structures of both systems. In this paper, we show that two chaotic systems linked through a feedback coupling interaction term of gain parameter k reach a synchronized regime characterized by a vector of variable errors which tends towards zero with parameter k while the interaction term tends towards a finite nonzero permanent regime. This means that maintaining a certain degree of synchronization has a cost. In the limit, complete synchronization occurs at a finite limit cost. We show that feedback coupling in itself brings about conditions permitting that systems with a degree of structural parameter flexibility evolve close towards each other structures in order to facilitate the maintenance of the synchronized regime. In this paper, we deduce parameter adaptive laws for any family of homochaotic systems provided they are previously forced to work, via feedback coupling, within an appropriate degree of synchronization. The laws are global in the space of parameters and lead eventually to identical synchronization at no interaction cost. We illustrate this point with homochaotic systems from the Lorenz, Rossler and Chua families.

Cost of synchronizing different chaotic systems

Feedback coupling provides a general scheme for synchronizing two oscillatory chaotic systems through the intervention of a term of interaction that accounts for the difference of behaviors. We define a cost of synchronization based on a measure of the interaction term. Synchronizing different systems is not cost free and the cost increases with the requirements imposed on the synchronized behavior. We prove that many systems can reach a regime of complete synchronization at a limited, and a priory computable, cost. For identical systems, the cost of complete synchronization is zero. Some different systems can also keep a completely synchronized behavior in some of their variables at zero cost. We propose to reserve the name identical synchronization for complete synchronization at zero cost. We compute the cost for different stages of synchronization between two systems as different as the Rossler and Lorenz systems and for homochaotic cases of both families. If the response system is flexible enough to adapt to the structure of the driving system, lower synchronization cost or, eventually, identical synchronization will be possible. In this paper, we deduce adaptation laws to reach identical synchronization for any family of homochaotic systems, and we illustrate their application for the Rossler and Lorenz cases.

Energy efficiency of information transmission by electrically coupled neurons.

The generation of spikes by neurons is energetically a costly process. This paper studies the consumption of energy and the information entropy in the signalling activity of a model neuron both when it is supposed isolated and when it is coupled to another neuron by an electrical synapse. The neuron has been modelled by a four-dimensional Hindmarsh-Rose type kinetic model for which an energy function has been deduced. For the isolated neuron values of energy consumption and information entropy at different signalling regimes have been computed. For two neurons coupled by a gap junction we have analyzed the roles of the membrane and synapse in the contribution of the energy that is required for their organized signalling. Computational results are provided for cases of identical and nonidentical neurons coupled by unidirectional and bidirectional gap junctions. One relevant result is that there are values of the coupling strength at which the organized signalling of two neurons induced by the gap junction takes place at relatively low values of energy consumption and the ratio of mutual information to energy consumption is relatively high. Therefore, communicating at these coupling values could be energetically the most efficient option.

Potentiodynamic passivation of iron in KOH solution. Application of the layer—pore resistance model

Abstract The layer—pore resistance model is applied to explain the potentiodynamic passivation of iron in 1 M KOH solution. The differential equation given by the model is numerically integrated and theoretical voltammograms are then simulated. Parameter in the model are calculated by fitting between theoretical and experimental results. This procedure is repeated for consecutive sweeps, and the evolution of the thickness of the passivating layer with cyclic sweeps is obtained.

ENERGY-LIKE FUNCTIONS FOR SOME DISSIPATIVE CHAOTIC SYSTEMS

Functions of the phase space variables that can considered as possible energy functions for a given family of dissipative chaotic systems are discussed. This kind of functions are interesting due to their use as an energy-like quantitative measure to characterize different aspects of dynamic behavior of associated chaotic systems. We have calculated quadratic energy-like functions for the cases of Lorenz, Chen, Lu–Chen and Chua, and show the patterns of dissipation of energy on their respective attractors. We also show that in the case of the Rossler system at least a fourth-order polynomial is required to properly represent its energy.

Nonzero error synchronization of chaotic systems via dynamic coupling

Abstract When feedback coupling is used to synchronize arbitrary chaotic systems large enough constant interaction gains lead to nearly complete synchronization at quasi-zero error. This forced oscillatory regime takes place in a region of phase space that, although natural for the guiding system, can result to be impracticable as an operating region for the guided system. However, we show that a dynamic feedback coupling with the appropriate variable gain can lead to a fully synchronized regime at a given nonzero synchronization error, that is, with the guided system operating on a desired region of the phase space. Computational results for oscillators of the Lorenz and Rossler families are shown. The cost of maintaining a couple of oscillatory Lorenz systems synchronized at different constant values of the synchronization error has been evaluated. To do so, an energy-like function associated to the state of the guided system has been defined.

Parameter adaptive global synchronization of Lorenz chaotic systems

A parameter adaptive rule that globally synchronizes oscillatory Lorenz chaotic systems with initially different parameter values is reported. The response system is defined according to a master-slave type of coupling that guarantees synchronization when parameters are identical. The parameters of the response system are then adapted to reach convergence to the drive parameters. The rule is very robust and works efficiently with different coupling schemes. Although, in general, it needs to have access to the three state variables of the drive system, if some information about the parameters is available it can be readapted to work less demandingly. For instance, we report here global synchronization that requires access to variable x only, when one parameter from the drive system is known.

Computing the Energy Cost of the Information Transmitted by Model Biological Neurons

We assign an energy function to a Hindmarsh‐Rose model of a neuron and use it to compute values of average energy consumption during its signalling activity. We also compute values of information entropy of an isolated neuron and of mutual information between two electrically coupled neurons. We find that for the isolated neuron the chaotic signaling regime is the one with the biggest ratio of information entropy to energy consumption. We also find that in the case of electrically coupled neurons there are values of the coupling strength at which the mutual information to energy consumption ratio is maximum, that is, that transmitting at that coupling conditions is energetically less expensive.

Synchronization of non-identical chaotic systems

Several schemes for synchronizing oscillatory chaotic systems have been studied. Phase synchronization happens to occur very easily even when different styles of systems are involved. Although complete synchronization between non-identical systems is in general more difficult to achieve this paper reports nearly complete synchronization between two Lorenz systems with different parameters and also between Lorenz and Rossler systems.