Transition from chimera/solitary states to traveling waves

We study numerically the spatiotemporal dynamics of a ring network of nonlocally coupled nonlinear oscillators, each represented by a two-dimensional discrete-time model of the classical van der Pol oscillator. It is shown that the discretized oscillator exhibits a richer behavior, combining the peculiarities of both the original system and its own dynamics. Moreover, a large variety of spatiotemporal structures is observed in the network of discrete van der Pol oscillators when the discretization parameter and the coupling strength are varied. Such regimes as the coexistence of multichimera state/traveling wave and solitary state are revealed for the first time and studied in detail. It is established that the majority of the observed chimera/solitary states, including the newly found ones, are transient towards the purely traveling wave mode. The peculiarities of the transition process and the lifetime (transient duration) of the chimera structures and the solitary state are analyzed depending on the system parameters, observation time, initial conditions, and influence of external noise

Influence of Sodium Inward Current on Dynamical Behaviour of Modified Morris-Lecar Model

This paper presents a modified Morris-Lecar model by incorporating the sodium inward current. The dynamical behaviour of the model in response to key parameters is investigated. The model exhibits various excitability properties as the values of parameters are varied. We have examined the effects of changes in maximum ion conductances and external current on the dynamics of the membrane potential. A detailed numerical bifurcation analysis is conducted. The bifurcation structures obtained in this study are not present in existing bifurcation studies of original Morris-Lecar model. The results in this study provides the interpretation of electrical activity in excitable cells and a platform for further study

Identification of single- and double-well coherence-incoherence patterns by the binary distance matrix

The study of chimera states or, more generally, coherence-incoherence patterns has led to the development of several tools for their identification and characterization. In this work, we extend the eigenvalue decomposition method to distinguish between single-well and double-well patterns. By applying our method, we are able to identify the following four types of dynamical patterns in a ring of nonlocally coupled Chua circuits and nonlocally coupled cubic maps: single-well cluster, single-well coherence-incoherence pattern, double-well cluster, and double-well coherence-incoherence. In a ring-star network of Chua circuits, we investigate the influence of adding a central node on the spatio-temporal patterns. Our results show that increasing the coupling with the central node favors the occurrence of single-well coherence-incoherence states. We observe that the boundaries of the attraction basins resemble fractal and riddled structure

Globally resonant homoclinic tangencies : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand

The attractors of a dynamical system govern its typical long-term behaviour. The presence of many attractors is significant as it means the behaviour is heavily dependent on the initial conditions. To understand how large numbers of attractors can coexist, in this thesis we study the occurrence of infinitely many stable single-round periodic solutions associated with homoclinic connections in two-dimensional maps. We show this phenomenon has a relatively high codimension requiring a homoclinic tangency and 'global resonance', as has been described previously in the area-preserving setting. However, unlike in that setting, local resonant terms also play an important role. To determine how the phenomenon may manifest in bifurcation diagrams, we also study perturbations of a globally resonant homoclinic tangency. We find there exist sequences of saddle-node and period-doubling bifurcations. Interestingly, in different directions of parameter space, the bifurcation values scale differently resulting in a complicated shape for the stability region for each periodic solution. In degenerate directions the bifurcation values scale substantially slower as illustrated in an abstract piecewise-smooth C¹ map

Dynamical effects of electromagnetic flux on Chialvo neuron map: nodal and network behaviors

We consider the dynamical effects of electromagnetic flux on the discrete Chialvo neuron. It is shown that the model can exhibit rich dynamical behaviors such as multistability, firing patterns, antimonotonicity, closed invariant curves, various routes to chaos, fingered chaotic attractors. The system enters chaos via period-doubling cascades, reverse period-doubling route, antimonotonicity, via closed invariant curve to chaos. The results were confirmed using the techniques of bifurcation diagrams, Lyapunov exponent diagram, phase portraits, basins of attraction and numerical continuation of bifurcations. Different global bifurcations are also shown to exist via numerical continuation. After understanding a single neuron model, a network of Chialvo neuron is explored. A ring-star network of Chialvo neuron is considered and different dynamical regimes such as synchronous, asynchronous, chimera states are revealed. Different continuous and piecewise continuous wavy patterns were also found during the simulations for negative coupling strengths.Comment: 35 pages, 22 figure

Repulsive inter-layer coupling induces anti-phase synchronization

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Shepelev, I. A., Muni, S. S., Schöll, E., & Strelkova, G. I. (2021). Repulsive inter-layer coupling induces anti-phase synchronization. In Chaos: An Interdisciplinary Journal of Nonlinear Science (Vol. 31, Issue 6, p. 063116). AIP Publishing. https://doi.org/10.1063/5.0054770 and may be found at https://doi.org/10.1063/5.0054770.We present numerical results for the synchronization phenomena in a bilayer network of repulsively coupled 2D lattices of van der Pol oscillators. We consider the cases when the network layers have either different or the same types of intra-layer coupling topology. When the layers are uncoupled, the lattice of van der Pol oscillators with a repulsive interaction typically demonstrates a labyrinth-like pattern, while the lattice with attractively coupled van der Pol oscillators shows a regular spiral wave structure. We reveal for the first time that repulsive inter-layer coupling leads to anti-phase synchronization of spatiotemporal structures for all considered combinations of intra-layer coupling. As a synchronization measure, we use the correlation coefficient between the symmetrical pairs of network nodes, which is always close to −1 in the case of anti-phase synchronization. We also study how the form of synchronous structures depends on the intra-layer coupling strengths when the repulsive inter-layer coupling is varied.DFG, 163436311, Kontrolle selbstorganisierender nichtlinearer Systeme: Theoretische Methoden und Anwendungskonzept