Nonlinear Sciences

Many living and non-living complex systems can be modeled and understood as collective systems made of heterogeneous components that self-organize and generate nontrivial morphological structures and behaviors. This chapter presents a brief overview of our recent effort that investigated various aspects of such morphogenetic collective systems. We first propose a theoretical classification scheme that distinguishes four complexity levels of morphogenetic collective systems based on the nature of their components and interactions. We conducted a series of computational experiments using a self-propelled particle swarm model to investigate the effects of (1) heterogeneity of components, (2) differentiation/re-differentiation of components, and (3) local information sharing among components, on the self-organization of a collective system. Results showed that (a) heterogeneity of components had a strong impact on the system's structure and behavior, (b) dynamic differentiation/re-differentiation of components and local information sharing helped the system maintain spatially adjacent, coherent organization, (c) dynamic differentiation/re-differentiation contributed to the development of more diverse structures and behaviors, and (d) stochastic re-differentiation of components naturally realized a self-repair capability of self-organizing morphologies. We also explored evolutionary methods to design novel self-organizing patterns, using interactive evolutionary computation and spontaneous evolution within an artificial ecosystem. These self-organizing patterns were found to be remarkably robust against dimensional changes from 2D to 3D, although evolution worked efficiently only in 2D settings.

In this paper, we numerically study the stochastic and the deterministic occasional uncoupling methods of effecting identical synchronized states in low dimensional, dissipative, diffusively coupled, chaotic flows that are otherwise not synchronized when continuously coupled at the same coupling strength parameter. In the process of our attempt to understand the mechanisms behind the success of the occasional uncoupling schemes, we devise a hybrid between the transient uncoupling and the stochastic on-off coupling, and aptly name it the transient stochastic uncoupling---yet another stochastic occasional uncoupling method. Our subsequent investigation on the transient stochastic uncoupling allows us to surpass the effectiveness of the stochastic on-off coupling with very fast on-off switching rate. Additionally, through the transient stochastic uncoupling, we establish that the indicators quantifying the local contracting dynamics in the corresponding transverse manifold are generally not useful in finding the optimal coupling region of the phase space in the case of the deterministic transient uncoupling. In fact, we highlight that the autocorrelation function---a non-local indicator of the dynamics---of the corresponding response system's chaotic time-series dictates when the deterministic uncoupling could be successful. We illustrate all our heuristic results using a few well-known examples of diffusively coupled chaotic oscillators.

Highly symmetric networks can exhibit partly symmetry-broken states, including clusters and chimera states, i.e., states of coexisting synchronized and unsynchronized elements. We address the S 4 permutation symmetry of four globally coupled Stuart-Landau oscillators and uncover an interconnected web of differently symmetric solutions. Among these are chaotic 2???? minimal chimeras that arise from 2???? periodic solutions in a period-doubling cascade, as well as fully asymmetric chaotic states arising similarly from periodic 1?????? solutions. A backbone of equivariant pitchfork bifurcations mediate between the two cascades, culminating in equivariant pitchforks of chaotic attractors.

The Kuramoto model, originally proposed to model the dynamics of many interacting oscillators, has been used and generalized for a wide range of applications involving the collective behavior of large heterogeneous groups of dynamical units whose states are characterized by a scalar angle variable. One such application in which we are interested is the alignment of orientation vectors among members of a swarm. Despite being commonly used for this purpose, the Kuramoto model can only describe swarms in 2 dimensions, and hence the results obtained do not apply to the often relevant situation of swarms in 3 dimensions. Partly based on this motivation, as well as on relevance to the classical, mean-field, zero-temperature Heisenberg model with quenched site disorder, in this paper we study the Kuramoto model generalized to D dimensions. We show that in the important case of 3 dimensions, as well as for any odd number of dimensions, the D -dimensional generalized Kuramoto model for heterogeneous units has dynamics that are remarkably different from the dynamics in 2 dimensions. In particular, for odd D the transition to coherence occurs discontinuously as the inter-unit coupling constant K is increased through zero, as opposed to the D=2 case (and, as we show, also the case of even D ) for which the transition to coherence occurs continuously as K increases through a positive critical value K c . We also demonstrate the qualitative applicability of our results to related models constructed specifically to capture swarming and flocking dynamics in three dimensions.

In this work we suggest to model the dynamics of power grids in terms of a two-layer network, and use the Italian high voltage power grid as a proof-of-principle example. The first layer in our model represents the power grid consisting of generators and consumers, while the second layer represents a dynamic communication network that serves as a controller of the first layer. In particular, the dynamics of the power grid is modelled by the Kuramoto model with inertia, while the communication layer provides a control signal P i for each generator to improve frequency synchronization within the power grid. We propose different realizations of the communication layer topology and different ways to calculate the control signal. Then we conduct a systematic survey of the two-layer system against a multitude of different realistic perturbation scenarios, such as disconnecting generators, increasing demand of consumers, or generators with stochastic power output. When using a control topology that allows all generators to exchange information, we find that a control scheme aimed to minimize the frequency difference between adjacent nodes operates very efficiently even against the worst scenarios with the strongest perturbations.

We consider a two-layer multiplex network of diffusively coupled FitzHugh-Nagumo (FHN) neurons in the excitable regime. It is shown, in contrast to SISR in a single isolated FHN neuron, that the maximum noise amplitude at which SISR occurs in the network of coupled FHN neurons is controllable, especially in the regime of strong coupling forces and long time delays. In order to use SISR in the first layer of the multiplex network to control CR in the second layer, we first choose the control parameters of the second layer in isolation such that in one case CR is poor and in another case, non-existent. It is then shown that a pronounced SISR cannot only significantly improve a poor CR, but can also induce a pronounced CR, which was non-existent in the isolated second layer. In contrast to strong intra-layer coupling forces, strong inter-layer coupling forces are found to enhance CR. While long inter-layer time delays just as long intra-layer time delays, deteriorates CR. Most importantly, we find that in a strong inter-layer coupling regime, SISR in the first layer performs better than CR in enhancing CR in the second layer. But in a weak inter-layer coupling regime, CR in the first layer performs better than SISR in enhancing CR in the second layer. Our results could find novel applications in noisy neural network dynamics and engineering.

We study the dynamics of two neuronal populations weakly and mutually coupled in a multiplexed ring configuration. We simulate the neuronal activity with the stochastic FitzHugh-Nagumo (FHN) model. The two neuronal populations perceive different levels of noise: one population exhibits spiking activity induced by supra-threshold noise (layer 1), while the other population is silent in the absence of inter-layer coupling because its own level of noise is sub-threshold (layer 2). We find that, for appropriate levels of noise in layer 1, weak inter-layer coupling can induce coherence resonance (CR), anti-coherence resonance (ACR) and inverse stochastic resonance (ISR) in layer 2. We also find that a small number of randomly distributed inter-layer links are sufficient to induce these phenomena in layer 2. Our results hold for small and large neuronal populations.

Pulse-coupled systems such as spiking neural networks exhibit nontrivial invariant sets in the form of attracting yet unstable saddle periodic orbits where units are synchronized into groups. Heteroclinic connections between such orbits may in principle support switching processes in those networks and enable novel kinds of neural computations. For small networks of coupled oscillators we here investigate under which conditions and how system symmetry enforces or forbids certain switching transitions that may be induced by perturbations. For networks of five oscillators we derive explicit transition rules that for two cluster symmetries deviate from those known from oscillators coupled continuously in time. A third symmetry yields heteroclinic networks that consist of sets of all unstable attractors with that symmetry and the connections between them. Our results indicate that pulse-coupled systems can reliably generate well-defined sets of complex spatiotemporal patterns that conform to specific transition rules. We briefly discuss possible implications for computation with spiking neural systems.

Convective self-aggregation is when thunderstorm clouds cluster over a constant temperature surface in radiative convective equilibrium. Self-aggregation was implicated in the Madden-Julian Oscillation and hurricanes. Yet, numerical simulations succeed or fail at producing self-aggregation, depending on modeling choices. Common explanations for self-aggregation invoke radiative effects, acting to concentrate moisture in a sub-domain. Interaction between cold pools, caused by rain evaporation, drives reorganization of boundary layer moisture and triggers new updrafts. We propose a simple model for aggregation by cold pool interaction, assuming a local number density ρ(r) of precipitation cells, and that interaction scales quadratically with ρ(r) . Our model mimics global energy constraints by limiting further cell production when many cells are present. The phase diagram shows a continuous phase transition between a continuum and an aggregated state. Strong cold pool-cold pool interaction gives a uniform convective phase, while weak interaction yields few and independent cells. Segregation results for intermediate interaction strength.

Recent works suggest that pooling and sharing may constitute a fundamental mechanism for the evolution of cooperation in well-mixed fluctuating environments. The rationale is that, by reducing the amplitude of fluctuations, pooling and sharing increases the steady-state growth rate at which the individuals self-reproduce. However, in reality interactions are seldom realized in a well-mixed structure, and the underlying topology is in general described by a complex network. Motivated by this observation, we investigate the role of the network structure on the cooperative dynamics in fluctuating environments, by developing a model for networked pooling and sharing of resources undergoing environmental fluctuations, represented through geometric Brownian motion. The study reveals that, while in general cooperation increases the individual steady state growth rates (i.e. is evolutionary advantageous), the interplay with the network structure may yield large discrepancies in the observed individual resource endowments. We comment possible biological and social implications and discuss relations to econophysics.