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PULSES, FRONTS AND CHAOTIC WAVE TRAINS IN A ONE-DIMENSIONAL CHUA’S LATTICE

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Papers International Journal of Bifurcation and Chaos Vol 7 No 8 1997 1775 1790 c World Scientific Publishing Company PULSES FRONTS AND CHAOTIC WAVE TRAINS IN A ONE DIMENSIONAL CHUA S LATTICE V B KAZANTSEV V I NEKORKIN The Nizhny Novgorod State University 23 Gagarin Ave 603600 Nizhny Novgorod Russia M G VELARDE Instituto Pluridisciplinar Universidad Complutense Paseo Juan XXIII N o 1 Madrid 28 040 Spain Received March 2 1997 Revised September 20 1997 We show how wave motions propagate in a nonequilibrium discrete medium modeled by a onedimensional array of diffusively coupled Chua s circuits The problem of the existence of the stationary wave solutions is reduced to the analysis of bounded trajectories of a fourth order system of nonlinear ODEs Then we study the homoclinic and heteroclinic bifurcations of the ODEs system The lattice can sustain the propagation of solitary pulses wave fronts and complex wave trains with periodic or chaotic profile 1 Introduction Many systems modeling processes in nonequilibrium excitable media are known to display solutions localized in space and steadily translating Solitary pulses fronts and wave trains propagating with a constant velocity are particular cases Examples come from all areas of science and engineering Take for instance waves in fluids Nepomnyashchy Velarde 1994 concentration waves in oscillatory reaction diffusion systems Zhabotinsky 1974 Kuramoto 1984 waves in optical fibers Hasegawa Kodama 1995 Huang Velarde 1996 pulses and pulse trains in long arrays of Josephson junctions Lonngren Scott 1995 transmission of excitation in neural fibers Murray 1993 etc Models allowing to describe such processes are appropriate Ginzburg Landau equations Van Saarloos Hohenberg 1992 the Korteweg de Vries equation Nekorkin Velarde 1994 Velarde et al 1995 Christov Velarde 1995 the model of Fitz Hugh Nagumo Murray 1993 and their generalizations Nepomnyashchy Velarde 1994 There has also been growing interest in models composed of coupled nonlinear oscillators located at the junctions or sites of a space lattice Such systems are also appropriate models for continuous media describing well for example the phenomena of pattern formation Nekorkin Chua 1993 wave propagation Perez Munuzuri et al 1993 Nekorkin et al 1995 1996 and spatially chaotic or spatio temporal chaotic processes Ogorzalek 1995 Nekorkin et al 1995 On the other hand lattice models naturally arise when using arrays of Josephson junctions arrays of reaction cells neural networks arrays of electronic oscillators etc whose dynamical behavior can be quite complex both in time and in space Perez Munuzuri et al 1993 Winfree 1991 Nekorkin et al 1995 Perez Marino et al 1995 Such lattices correspond to discrete nonequilibrium media Finally such models can be used suitably for simulation of a given natural process by means of for example appropriate electronic circuits which would exhibit a required collective behavior This is of great importance both from the applied viewpoint and for the understanding of dynamical processes in nature We study the dynamics of a one dimensional array of diffusively coupled Chua s circuits Chua 1775 1776 V B Kazantsev et al 1993 Madan 1993 In recent studies such array has been used for modeling biological fibers neural networks reaction diffusion systems etc It has been found that the array can exhibit pattern formation or spatial disorder propagation of wave fronts reentry initiation of pulses in two coupled arrays spiral and scroll waves in two dimensional arrays etc The possibility of traveling pulses and wave trains has been shown in Nekorkin et al 1995 for the case of inductive coupling between cells However generally these solutions although may be long lasting structures they are not stable In this paper we discuss this problem for the case of resistive diffusive coupling and show how the array can be considered as an excitable fiber capable of sustaining the propagation of various types of stable travelling waves including single pulses or fronts and complex wave trains with a periodic or a chaotic sequence of pulses The profiles of possible travelling waves are derived as bounded trajectories of the fourth order system of ODEs underlying the original space dependent problem states of the array These states are O xj yj zj 0 P xj x0 yj y0 zj z0 P xj x0 yj y0 zj z0 where x0 b a b b 1 y0 b a b b 1 z0 b a b b 1 It has been already shown in Nekorkin et al 1993 that for each set of the parameter values the outer states P and P are locally asymptotically stable while the trivial state O is unstable Thus the array is a discrete medium with two excitable steady states We now show that this medium is able to sustain stable localized solutions fronts pulses pulse trains travelling in space with definite velocity 2 Model The dynamics of 1 D lattice of diffusively coupled Chua s circuits can be described in dimensionless form by the following set of ODEs see e g Nekorkin Chua 1993 x j yj xj f xj d xj 1 2xj xj 1 3 Possible Profiles of Travelling Waves Here we prove the existence of travelling wave solutions in the system 1 and determine some characteristics of the waves profile form velocity of propagation y j xj yj zj 3 1 Travelling waves z j yj zj j 1 2 N 1 where f x describes the symmetric three segment piecewise linear function bx a b f x if 1 x 1 bx a b if x 1 xj t x yj t y if x 1 ax Let us look for a solution of the system 1 in the form of a travelling wave We pose 3 zj t z 2 with a 0 and b 0 The other parameters of the system d are also taken positive We shall consider two types of boundary conditions i zero flux conditions x0 x1 xN 1 xN and ii periodic conditions x0 xN xN 1 x1 The latter describes a circular array The system 1 has three equilibria or fixed points corresponding to the homogeneous steady where t jh is a coordinate moving along the array with a constant velocity c 1 h Thus x y z describes a space profile steadily translating with the velocity c For solutions of the form 3 the system 1 is reduced to x y x f x d x h 2x x h y x y z z y z 4 Pulses Fronts and Chaotic Wave Trains in a One Dimensional Chua s Lattice 1777 where the dot denotes the differentiation with respect to the moving coordinate In the continuum approximation see e g Nekorkin et al 1995 i e when the spatial grid of the solution profile is significantly finer than the spatial grid of the discrete array it is possible to change the difference term in Eq 4 by using the second derivative x


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