Articles written in Indian Academy of Sciences Conference Series
Volume 1 Issue 1 December 2017 pp v-vi Proceedings of the Conference on Perspectives in Nonlinear Dynamics - 2016
Volume 1 Issue 1 December 2017 pp 77-83 Proceedings of the Conference on Perspectives in Nonlinear Dynamics - 2016
Mapping time series onto graphs and the use of graph theory methods opens up the possibility to study the structure of the phase space manifolds underlying the fluctuations of a dynamical variable. Here, we go beyond the standard graph measures and analyze the higher-order structures such as triangles, tetrahedra and higher-order cliques and their complexes present in the time-series networks, which are detectable by algebraic topology methods. We investigate the Barkhausen noise signal which accompanies domain-wall dynamics during magnetization reversal in weakly disordered ferromagnets by a slow increase of the external field along the hysteresis loop. Our analysis demonstrates how the appearance of the complexes with cliques of a high order correlates to the enhanced collective fluctuations in the central part of the hysteresis loop, where domain-wall depinning occurs. In contrast, the fractional Gaussian noise fluctuations at the beginning of the loop correspond to the graph of a simpler topology. The determined topology measures serve as geometrical indicators of changing dynamical regimes along the hysteresis loop. The multifractal analysis of the corresponding segments of the signal confirms that we deal with different types of stochastic processes.
Volume 1 Issue 1 December 2017 pp 195-203 Proceedings of the Conference on Perspectives in Nonlinear Dynamics - 2016
We show the existence and stability of frozen splay states as well as temporally chaotic splay states in a coupled sine circle map lattice system using analytic and numerical techniques. The splay states are observed for very low values of the nonlinearity parameter, i.e., for circle maps which deviate very slightly from the shift map case. We also observe that, depending on the parameters of the system, the splay states bifurcate to mixed or chimera splay states, consisting of a mixture of splay and synchronised states, together with kinks in the phases of some of the maps and then to a globally synchronised state. We estimate the parameter regions where these pure states and the mixed states are seen. We also briefly show that similar spatial splay structures can exist in experimentally realisable systems like Josephson junction arrays and Hartley-like oscillator arrays.
Volume 1 Issue 1 December 2017 pp 205-212 Proceedings of the Conference on Perspectives in Nonlinear Dynamics - 2016
The problem of synchronization of coupled Hamiltonian systems exhibits interesting features due to the non-uniform or mixed nature (regular and chaotic) of the phase space. We study these features by investigating the synchronization of unidirectionally coupled area-preserving maps coupled by the Pecora–Carroll method. We find that coupled standard maps show complete synchronization for values of the nonlinearity parameter at which regular structures are still present in phase space. The distribution of synchronization times has a power law tail indicating long synchronization times for at least some of the synchronizing trajectories. With the introduction of coherent structures using parameter perturbation in the system, this distribution crosses over to exponential behavior, indicating shorter synchronization times, and the number of initial conditions which synchronize increases significantly, indicating an enhancement in the basin of synchronization. On the other hand, coupled blinking vortex maps display both phase synchronization and phase slips, depending on the location of the initial conditions. We discuss the implication of our results.
Volume 2 All articles Published: 16 September 2019 Article ID 0022 Special
The prediction of the critical point of a phase transition is a topic of great current interest, and is of utility in many practical contexts. Therefore, the identification of precursors, or early warning signals of the critical point, has become the focus of current interest. Recent model studies have shown that a series of small transitions, which have been called microtransitions, act as precursors to the percolation transition. Here, we identify the existence of microtransitions in two distinct networks, for two distinct processes. The first case is the process of avalanche transmission on branching hierarchical networks. Here, typical realizations of the original lattice of this network exhibit a second order transition.We note that microtransitions in the variance of the order parameter are seen in this case. Additionally, the positions of the microtransitions follow ascaling relation. The scaling relation can be used to calculate the position of the critical point, which is seen to be in agreement with the observed result.We also introduce this method of identifying the microtransitions occurring before the tipping point to a complex real world system, the climate system. We analyse the discontinuous first order phase transition occurring in the climate networks. We apply the percolation framework to these networks to analyse the structural changes in the network and construct an order parameter and a susceptibility. Microtransitions can be found in the behaviour of the susceptibility. These can be used to predict the tipping point in the system. We discuss possible applicationsof this.