Climate network analysis of extreme events: Tropical cyclones.

Climate networks based on surface air temperature data are analyzed to identify distinct signatures of tropical cyclones, which appear in the Indian Ocean. These networks, which are percolating networks, show an abrupt phase transition in the order parameter and the susceptibility during cyclonic events. The behavior seen is compared for the months October-November 2016, when three successive cyclones, viz., cyclone Kyant, cyclone Nada, and cyclone Vardah, were seen, and compared with a year where a single cyclone, cyclone Ockhi, was seen in December 2017. All these cyclones were seen in the Bay of Bengal. The microtransitions, i.e., the locations of jumps in the order parameter, for these two cases show distinct patterns. The signatures of the cyclones can be seen in other quantities like the node degrees and their geographic distributions and other network characterizers. We also compare these with a cyclone, cyclone Ashoba (2015), seen in the Arabian Sea where cyclones are rarer. The networks also show the signatures of precursor behavior, which has implications for further analysis.

Robustness of the emergence of synchronized clusters in branching hierarchical systems under parametric noise.

The dynamical robustness of networks in the presence of noise is of utmost fundamental and applied interest. In this work, we explore the effect of parametric noise on the emergence of synchronized clusters in diffusively coupled Chaté-Manneville maps on a branching hierarchical structure. We consider both quenched and dynamically varying parametric noise. We find that the transition to a synchronized fixed point on the maximal cluster is robust in the presence of both types of noise. We see that the small sub-maximal clusters of the system, which coexist with the maximal cluster, exhibit a power-law cluster size distribution. This power-law scaling of synchronized cluster sizes is robust against noise in a broad range of coupling strengths. However, interestingly, we find a window of coupling strength where the system displays markedly different sensitivities to noise for the maximal cluster and the small clusters, with the scaling exponent for the cluster distribution for small clusters exhibiting clear dependence on noise strength, while the cluster size of the maximal cluster of the system displays no significant change in the presence of noise. Our results have implications for the observability of synchronized cluster distributions in real-world hierarchical networks, such as neural networks, power grids, and communication networks, that necessarily have parametric fluctuations.

Effect of hidden geometry and higher-order interactions on the synchronization and hysteresis behavior of phase oscillators on 5-clique simplicial assemblies.

The hidden geometry of simplicial complexes can influence the collective dynamics of nodes in different ways depending on the simplex-based interactions of various orders and competition between local and global structural features. We study a system of phase oscillators attached to nodes of four-dimensional simplicial complexes and interacting via positive/negative edges-based pairwise K_{1} and triangle-based triple K_{2}≥0 couplings. Three prototypal simplicial complexes are grown by aggregation of 5-cliques, controlled by the chemical affinity parameter ν, resulting in sparse, mixed, and compact architecture, all of which have 1-hyperbolic graphs but different spectral dimensions. By changing the interaction strength K_{1}∈[-4,2] along the forward and backward sweeps, we numerically determine individual phases of each oscillator and a global order parameter to measure the level of synchronization. Our results reveal how different architectures of simplicial complexes, in conjunction with the interactions and internal-frequency distributions, impact the shape of the hysteresis loop and lead to patterns of locally synchronized groups that hinder global network synchronization. Remarkably, these groups are differently affected by the size of the shared faces between neighboring 5-cliques and the presence of higher-order interactions. At K_{1}<0, partial synchronization is much higher in the compact community than in the assemblies of cliques sharing single nodes, at least occasionally. These structures also partially desynchronize at a lower triangle-based coupling K_{2} than the compact assembly. Broadening of the internal frequency distribution gradually reduces the synchronization level in the mixed and sparse communities, even at positive pairwise couplings. The order-parameter fluctuations in these partially synchronized states are quasicyclical with higher harmonics, described by multifractal analysis and broad singularity spectra.

Signatures of climatic phenomena in climate networks: El Niño and La Niña.

We construct climate networks based on surface air temperature data to identify distinct signatures of climatic phenomena such as El Niño and La Niña events, which trigger many climatic disruptions around the globe with severe economic and ecological consequences. The climate network has been seen to show a discontinuous phase transition in the size of the normalized largest cluster and the susceptibility during both events. The correlation matrix of the network shows a structure that has distinct characteristics for El Niño events, La Niña events, and normal conditions of the Pacific Ocean. We also identify the signatures of the El Niño southern oscillations in the heat map of the cross-correlation and network quantifiers like the betweenness centrality. The distribution of teleconnections of distinct strengths, the betweenness centrality distributions, and the geographic location of nodes of high betweenness centrality yield important insights into the structure of the network and the transfer of information between different parts. We further discuss the predictive power of these quantities.

The transition to synchronization on branching hierarchical lattices.

We study the transition to synchronization on hierarchical lattices using the evolution of Chaté-Manneville maps placed on a triangular lattice. Connections are generated between the levels of the triangular lattice, assuming that each site is connected to its neighbors on the level below with probability half. The maps are diffusively coupled, and the map parameters increase hierarchically, depending on the map parameters at the sites they are coupled to in the previous level. The system shows a transition to synchronization, which is second order in nature, with associated critical exponents. However, the V-lattice, which is a special realization of this lattice, shows a transition to synchronization that is discontinuous with accompanying hysteretic behavior. This transition can thus be said to belong to the class of explosive synchronization with the explosive nature depending on the nature of the substrate. We carry out finite-size-finite-time scaling for the continuous transition and analyze the scaling of the jump size for the discontinuous case. We discuss the implications of our results and draw parallels with avalanche statistics on branching hierarchical lattices.

Multifractal analysis of birdsong and its correlation structure.

The time series recordings of typical songs of songbirds exhibit highly complex and structured behavior, which is characteristic of their species and stage of development, and need to be analyzed by methods that can uncover their correlation structure. Here we analyze a typical song of a canary using Hurst exponents and multifractal analysis, which uncovers the correlation structure of typical song segments. These are then compared with the corresponding quantities from shuffled data, which destroys the temporal correlations and iterative amplitude-adjusted Fourier transform (IAAFT) data. It is seen that temporal correlations are responsible for the multifractal behavior seen in the data and that two-point correlations, which are preserved by the transform, are important in the high-fluctuation regime. Higher-order correlations and intersyllabic gaps dominate the behavior of the low-fluctuation regime. These observations are supported by the simplicial characterization of the corresponding time series networks. Complexity measures are also used to analyze the amplitude envelope time series. These indicate that intersyllabic gaps contribute a significant fraction to the complexity of the birdsong. Our method provides a detailed characterization of the data, which can enable the comparison of real and synthetic birdsong and comparisons across stages of development and species. A brief comparison with the song of the zebra finch supports this.

A simplicial analysis of the fMRI data from human brain dynamics under functional cognitive tasks.

The topological analysis of fMRI time series data has recently been used to characterize the identification of patterns of brain activity seen during specific tasks carried out under experimentally controlled conditions. This study uses the methods of algebraic topology to characterize time series networks constructed from fMRI data measured for adult and children populations carrying out differentiated reading tasks. Our pilot study shows that our methods turn out to be capable of identifying distinct differences between the activity of adult and children populations carrying out identical reading tasks. We also see differences between activity patterns seen when subjects recognize word and nonword patterns. The results generalize across different populations, different languages and different active and inactive brain regions.

Hysteresis and synchronization processes of Kuramoto oscillators on high-dimensional simplicial complexes with competing simplex-encoded couplings.

Recent studies of dynamic properties in complex systems point out the profound impact of hidden geometry features known as simplicial complexes, which enable geometrically conditioned many-body interactions. Studies of collective behaviors on the controlled-structure complexes can reveal the subtle interplay of geometry and dynamics. Here we investigate the phase synchronization (Kuramoto) dynamics under the competing interactions embedded on 1-simplex (edges) and 2-simplex (triangles) faces of a homogeneous four-dimensional simplicial complex. Its underlying network is a 1-hyperbolic graph with the assortative correlations among the node's degrees and the spectral dimension that exceeds d_{s}=4. By numerically solving the set of coupled equations for the phase oscillators associated with the network nodes, we determine the time-averaged system's order parameter to characterize the synchronization level. Our results reveal a variety of synchronization and desynchronization scenarios, including partially synchronized states and nonsymmetrical hysteresis loops, depending on the sign and strength of the pairwise interactions and the geometric frustrations promoted by couplings on triangle faces. For substantial triangle-based interactions, the frustration effects prevail, preventing the complete synchronization and the abrupt desynchronization transition disappears. These findings shed new light on the mechanisms by which the high-dimensional simplicial complexes in natural systems, such as human connectomes, can modulate their native synchronization processes.

Precursors of the El Niño phenomenon: A climate network analysis.

The identification of precursors of climatic phenomena has enormous practical importance. Recent work constructs a climate network based on surface air temperature data to analyze the El Niño phenomena. We utilize microtransitions which occur before the discontinuous percolation transition in the network as well as other network quantities to identify a set of reliable precursors of El Niño episodes. These precursors identify 10 out of 13 El Niño episodes occurring in the period of 1979-2018 with an average lead time of approximately 6.4 months. We also find indicators of tipping events in the data.

Phase synchronization in the two-dimensional Kuramoto model: Vortices and duality.

We study a system of Kuramoto oscillators arranged on a two-dimensional periodic lattice where the oscillators interact with their nearest neighbors, and all oscillators have the same natural frequency. The initial phases of the oscillators are chosen to be distributed uniformly between (-π,π]. During the relaxation process to the final stationary phase, we observe different features in the phase field of the oscillators: initially, the state is randomly oriented, then clusters form. As time evolves, the size of the clusters increases and vortices that constitute topological defects in the phase field form in the system. These defects, being topological, annihilate in pairs; i.e., a given defect annihilates if it encounters another defect with opposite polarity. Finally, the system ends up either in a completely phase synchronized state in case of complete annihilation or a metastable phase locked state characterized by presence of vortices and antivortices. The basin volumes of the two scenarios are estimated. Finally, we carry out a duality transformation similar to that carried out for the XY model of planar spins on the Hamiltonian version of the Kuramoto model to expose the underlying vortex structure.

Chimera states in coupled map lattices: Spatiotemporally intermittent behavior and an equivalent cellular automaton.

We construct an equivalent cellular automaton (CA) for a system of globally coupled sine circle maps with two populations and distinct values for intergroup and intragroup coupling. The phase diagram of the system shows that the coupled map lattice can exhibit chimera states with synchronized and spatiotemporally intermittent subgroups after evolution from random initial conditions in some parameter regimes, as well as to other kinds of solutions in other parameter regimes. The CA constructed by us reflects the global nature and the two population structure of the coupled map lattice and is able to reproduce the phase diagram accurately. The CA depends only on the total number of laminar and burst sites and shows a transition from co-existing deterministic and probabilistic behavior in the chimera region to fully probabilistic behavior at the phase boundaries. This identifies the characteristic signature of the transition of a cellular automaton to a chimera state. We also construct an evolution equation for the average number of laminar/burst sites from the CA, analyze its behavior and solutions, and correlate these with the behavior seen for the coupled map lattice. Our CA and methods of analysis can have relevance in wider contexts.

Characterizing the complexity of time series networks of dynamical systems: A simplicial approach.

We analyze the time series obtained from different dynamical regimes of evolving maps and flows by constructing their equivalent time series networks, using the visibility algorithm. The regimes analyzed include periodic, chaotic, and hyperchaotic regimes, as well as intermittent regimes and regimes at the edge of chaos. We use the methods of algebraic topology, in particular, simplicial complexes, to define simplicial characterizers, which can analyze the simplicial structure of the networks at both the global and local levels. The simplicial characterizers bring out the hierarchical levels of complexity at various topological levels. These hierarchical levels of complexity find the skeleton of the local dynamics embedded in the network, which influence the global dynamical properties of the system and also permit the identification of dominant motifs. We also analyze the same networks using conventional network characterizers such as average path lengths and clustering coefficients. We see that the simplicial characterizers are capable of distinguishing between different dynamical regimes and can pick up subtle differences in dynamical behavior, whereas the usual characterizers provide a coarser characterization. However, the two taken in conjunction can provide information about the dynamical behavior of the time series, as well as the correlations in the evolving system. Our methods can, therefore, provide powerful tools for the analysis of dynamical systems.

Transport, diffusion, and energy studies in the Arnold-Beltrami-Childress map.

We study the transport and diffusion properties of passive inertial particles described by a six-dimensional dissipative bailout embedding map. The base map chosen for the study is the three-dimensional incompressible Arnold-Beltrami-Childress (ABC) map chosen as a representation of volume preserving flows. There are two distinct cases: the two-action and the one-action cases, depending on whether two or one of the parameters (A,B,C) exceed 1. The embedded map dynamics is governed by two parameters (α,γ), which quantify the mass density ratio and dissipation, respectively. There are important differences between the aerosol (α<1) and the bubble (α>1) regimes. We have studied the diffusive behavior of the system and constructed the phase diagram in the parameter space by computing the diffusion exponents η. Three classes have been broadly classified-subdiffusive transport (η<1), normal diffusion (η≈1), and superdiffusion (η>1) with η≈2 referred to as the ballistic regime. Correlating the diffusive phase diagram with the phase diagram for dynamical regimes seen earlier, we find that the hyperchaotic bubble regime is largely correlated with normal and superdiffusive behavior. In contrast, in the aerosol regime, ballistic superdiffusion is seen in regions that largely show periodic dynamical behaviors, whereas subdiffusive behavior is seen in both periodic and chaotic regimes. The probability distributions of the diffusion exponents show power-law scaling for both aerosol and bubbles in the superdiffusive regimes. We further study the Poincáre recurrence times statistics of the system. Here, we find that recurrence time distributions show power law regimes due to the existence of partial barriers to transport in the phase space. Moreover, the plot of average particle kinetic energies versus the mass density ratio for the two-action case exhibits a devil's staircase-like structure for higher dissipation values. We explain these results and discuss their implications for realistic systems.

Spatial splay states and splay chimera states in coupled map lattices.

We study the existence and stability of splay states in the 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 maps which deviate very slightly from the shift map case. We also observe that depending on the parameters of the system the splay state bifurcates to a mixed or chimera splay state consisting of a mixture of splay and synchronized states, together with kinks in the phases of some of the maps and then to a stable globally synchronized state. We show that these pure states and the mixed states are all temporally chaotic for our systems, and we explore the stability of these states to perturbations. Our studies may provide pointers to the behavior of systems in diverse application contexts such as Josephson junction arrays and chemical oscillations.

Synchronization in area-preserving maps: Effects of mixed phase space and coherent structures.

The problem of synchronization of coupled Hamiltonian systems presents interesting features due to the mixed nature (regular and chaotic) of the phase space. We study these features by examining the synchronization of unidirectionally coupled area-preserving maps coupled by the Pecora-Caroll method. The master stability function approach is used to study the stability of the synchronous state and to identify the percentage of synchronizing initial conditions. The transient to synchronization shows intermittency with an associated power law. The mixed nature of the phase space of the studied map has notable effects on the synchronization times as is seen in the case of the standard map. Using finite-time Lyapunov exponent analysis, we show that the synchronization of the maps occurs in the neighborhood of invariant curves in the phase space. The phase differences of the coevolving trajectories show intermittency effects, due to the existence of stable periodic orbits contributing locally stable directions in the synchronizing neighborhoods. Furthermore, the value of the nonlinearity parameter, as well as the location of the initial conditions play an important role in the distribution of synchronization times. We examine drive response combinations which are chaotic-chaotic, chaotic-regular, regular-chaotic, and regular-regular. A range of scaling behavior is seen for these cases, including situations where the distributions show a power-law tail, indicating long synchronization times for at least some of the synchronizing trajectories. The introduction of coherent structures in the system changes the situation drastically. The distribution of synchronization times 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. We discuss the implications of our results.

Hidden geometry of traffic jamming.

We introduce an approach based on algebraic topological methods that allow an accurate characterization of jamming in dynamical systems with queues. As a prototype system, we analyze the traffic of information packets with navigation and queuing at nodes on a network substrate in distinct dynamical regimes. A temporal sequence of traffic density fluctuations is mapped onto a mathematical graph in which each vertex denotes one dynamical state of the system. The coupling complexity between these states is revealed by classifying agglomerates of high-dimensional cliques that are intermingled at different topological levels and quantified by a set of geometrical and entropy measures. The free-flow, jamming, and congested traffic regimes result in graphs of different structure, while the largest geometrical complexity and minimum entropy mark the edge of the jamming region.

Dynamics of impurities in a three-dimensional volume-preserving map.

We study the dynamics of inertial particles in three-dimensional incompressible maps, as representations of volume-preserving flows. The impurity dynamics has been modeled, in the Lagrangian framework, by a six-dimensional dissipative bailout embedding map. The fluid-parcel dynamics of the base map is embedded in the particle dynamics governed by the map. The base map considered for the present study is the Arnold-Beltrami-Childress (ABC) map. We consider the behavior of the system both in the aerosol regime, where the density of the particle is larger than that of the base flow, as well as the bubble regime, where the particle density is less than that of the base flow. The phase spaces in both the regimes show rich and complex dynamics with three types of dynamical behaviors--chaotic structures, regular orbits, and hyperchaotic regions. In the one-action case, the aerosol regime is found to have periodic attractors for certain values of the dissipation and inertia parameters. For the aerosol regime of the two-action ABC map, an attractor merging and widening crisis is identified using the bifurcation diagram and the spectrum of Lyapunov exponents. After the crisis an attractor with two parts is seen, and trajectories hop between these parts with period 2. The bubble regime of the embedded map shows strong hyperchaotic regions as well as crisis induced intermittency with characteristic times between bursts that scale as a power law behavior as a function of the dissipation parameter. Furthermore, we observe a riddled basin of attraction and unstable dimension variability in the phase space in the bubble regime. The bubble regime in the one-action case shows similar behavior. This study of a simple model of impurity dynamics may shed light upon the transport properties of passive scalars in three-dimensional flows. We also compare our results with those seen earlier in two-dimensional flows.

Crisis, unstable dimension variability, and bifurcations in a system with high-dimensional phase space: coupled sine circle maps.

The phenomenon of crisis in systems evolving in high-dimensional phase space can show unexpected and interesting features. We study this phenomenon in the context of a system of coupled sine circle maps. We establish that the origins of this crisis lie in a tangent bifurcation in high dimensions, and identify the routes that lead to the crisis. Interestingly, multiple routes to crisis are seen depending on the initial conditions of the system, due to the high dimensionality of the space in which the system evolves. The statistical behavior seen in the phase diagram of the system is also seen to change due to the dynamical phenomenon of crisis, which leads to transitions from nonspreading to spreading behavior across an infection line in the phase diagram. Unstable dimension variability is seen in the neighborhood of the infection line. We characterize this crisis and unstable dimension variability using dynamical characterizers, such as finite-time Lyapunov exponents and their distributions. The phase diagram also contains regimes of spatiotemporal intermittency and spatial intermittency, where the statistical quantities scale as power laws. We discuss the signatures of these regimes in the dynamic characterizers, and correlate them with the statistical characterizers and bifurcation behavior. We find that it is necessary to look at both types of correlators together to build up an accurate picture of the behavior of the system.

Transmission of packets on a hierarchical network: statistics and explosive percolation.

We analyze an idealized model for the transmission or flow of particles, or discrete packets of information, in a weight bearing branching hierarchical two dimensional network and its variants. The capacities add hierarchically down the clusters. Each node can accommodate a limited number of packets, depending on its capacity, and the packets hop from node to node, following the links between the nodes. The statistical properties of this system are given by the Maxwell-Boltzmann distribution. We obtain analytical expressions for the mean occupation numbers as functions of capacity, for different network topologies. The analytical results are shown to be in agreement with the numerical simulations. The traffic flow in these models can be represented by the site percolation problem. It is seen that the percolation transitions in the 2D model and in its variant lattices are continuous transitions, whereas the transition is found to be explosive (discontinuous) for the V lattice, the critical case of the 2D lattice. The scaling behavior of the second-order percolation case is studied in detail. We discuss the implications of our analysis.

On the neural substrates for exploratory dynamics in basal ganglia: a model.

We present a neural network model of basal ganglia that departs from the classical Go/NoGo picture of the function of its key pathways-the direct pathway (DP) and the indirect pathway (IP). In classical descriptions of basal ganglia function, the DP is known as the Go pathway since it facilitates movement and the IP is called the NoGo pathway since it inhibits movement. Between these two regimes, in the present model, we posit that there is a third Explore regime, which denotes random exploration of the space of actions. The proposed model is instantiated in a simple action selection task. Striatal dopamine is assumed to switch between DP and IP activation. The IP is modeled as a loop of the subthalamic nucleus (STN) and the globus pallidus externa (GPe), capable of producing chaotic activity. Simulations reveal that, while the system displays Go and NoGo regimes for extreme values of dopamine, at intermediate values of dopamine, it exhibits a new Explore regime denoting a random exploration of the space of action alternatives. The exploratory dynamics originates from the chaotic activity of the STN-GPe loop. When applied to the standard card choice experiment used in the imaging studies of Daw, O'Doherty, Dayan, Seymour, and Dolan (2006), the model favorably describes the exploratory behavior of human subjects.