Influence of noise on spiral and target wave regimes in two-dimensional lattice of locally coupled maps

E. V. Rybalova ; V. S. Anishchenko

Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 29 (2021) no. 2, pp. 272-287.

Résumé

The objective is to study numerically the dynamics of two-dimensional lattice of locally coupled maps of Rulkov. We analyze conditions for the appearance and existence as well as the properties of auto-wave spatio-temporal structures which are represented by spiral and target waves. The influence of noise on the lattice dynamics is explored as the noise intensity and the size of the noise-disturbed region are varied. Methods. In numerical experiments the evolution of the lattice dynamics is directly determined by the corresponding recurrence relations. The numerical data are used to construct spatial distributions of the instantaneous values of the amplitudes for all the network elements, spatio-temporal diagrams for the lattice cross-section at different values of the control parameters of the individual nodes, for various noise intensities and different sizes of the noise-disturbed region. The obtained results are compared. The noise-disturbed region is specified as a square which consists of a small number of oscillators at the lattice center. Results. It is found that for certain values of the control parameters of the maps, of the coupling parameters, and the initial conditions, long-lived spiral and target waves can exist in the lattice. It is shown that the spiral wave regimes are, as a rule, transient, can be observed for a finite time and become long-lived only for certain values of the parameters and the initial conditions. When the noise influences a finite region of the lattice showing spiral waves, the transition to spiral waves with a different structure or to target waves can occur. However, if the noise disturbance is removed, the lattice returns to its original mode or exhibits the transition to coherent dynamics modes. The target waves are more resistant to the noise and are observed for longer times. If the noise causes the target waves to change, the resulting regime continues to exist after removing the noise source. Conclusion. It is shown that the spiral and target waves can be observed in the lattice of locally coupled Rulkov maps. The regions where these waves exist are defined and constructed in the plane of the control parameters of the individual elements. Studying the impact of the relation between the noise intensity and the size of the noise-disturbed region enables one to distinguish the region where the transition from spiral to target waves always occurs, as well as the area inside which this transition depends on the initial states of the lattice elements and the noise realization. The effect of noise on the target waves can induce the appearance of only target wave chimeras which continue to exist even after the noise excitation is turned off.

Détail

Keywords: two-dimensional ensemble, locally coupling, spiral wave, target wave, noise influence.