Inverse Network Dynamics - Network structure and function from nonlinear dynamics and time series data

When chaotic systems are coupled to each other through state variables, synchronization can happen. Here, we present an analytical study for the reconstruction of chaotic synchronized networks based on average integrated causation entropies. We show that if we inject the random information via impulsive perturbations into the individual systems to destroy the synchronization briefly, that can lead to the prediction of the network structure en route to re-synchronization. An algorithm based on theoretical results is developed to reconstruct the adjacency matrix of the network. The results indicate that the proposed algorithm can be used to reconstruct the network en route to re-synchronization.

In this presentation, we explore a few distinct types of intermittent large-intensity pulses, namely, Pomeau-Manneville intermittency, quasiperiodic intermittency, and quasiperiodic breakdown to chaos, that originate from periodic, quasiperiodic, and chaotic states, respectively, in Zeeman laser. During the transitions to large-intensity pulses, we observe the origin of hyperchaos. We classify the transitions to hyperchaos as discontinuous against a parameter change, in all the cases, however, from periodic to hyperchaos, it shows hysteresis, but during quasiperiodic and chaotic to hyperchaos transition, no hysteresis is recorded. These features are revealed by plotting the two largest Lyapunov exponents of the laser model against the parameter. The turbulent phases of intermittency always consist of large intensity pulses. The large intensity pulses show characteristic features of extremes, in the sense, that they are larger than a significant height and have a probability of rare occurrence.

The ordinal pattern-based Complexity-Entropy Plane is a popular tool in nonlinear dynamics for distinguishing noise from chaos. While successful attempts to do so have been documented for low-dimensional maps and continuous-time systems, high-dimensional systems have not been investigated as thoroughly. To address the question in which way time series from systems governed by high-dimensional chaos can be characterized by their location in the Complexity-Entropy Plane we analyze data from several high-dimensional systems of different types, including spatially extended systems, dynamical networks and delay systems. We demonstrate the importance of the choice of the length and lag of the patterns, as well as the influence of the available data length and attractor dimension and discuss their impact on surrogate data tests.

The phase sensitivity curve or phase response curve (PRC) quantifies the oscillator's reaction to stimulation at a specific phase and is a primary characteristic of a self-sustained oscillatory unit. Knowledge of this curve yields a phase dynamics description of the oscillator for arbitrary weak forcing. Similar, though much less studied characteristic, is the amplitude response that can be defined either using an ad hoc approach to amplitude estimation or via the isostable variables. Here, we discuss the problem of the phase and amplitude response inference from observations using test stimulation. Although PRC determination for noise-free neuronal-like oscillators perturbed by narrow pulses is a well-known task, the general case remains a challenging problem. Even more challenging is the inference of the amplitude response. This characteristic is crucial, e.g., for controlling the amplitude of the collective mode in a network of interacting units; this task is relevant for neuroscience. Here, we compare the performance of different techniques suitable for inferring the phase and amplitude response, particularly with application to macroscopic oscillators. We suggest improvements to these techniques, e.g., demonstrating how to obtain the PRC in case of stimuli of arbitrary shape. Our main result is a novel technique denoted by IPID-1, based on the direct reconstruction of the Winfree equation and the analogous first-order equation for isostable dynamics. The technique works for signals with or without well-pronounced marker events and pulses of arbitrary shape; in particular, we consider charge-balanced pulses typical in neuroscience applications. Moreover, this technique is superior for noisy and high-dimensional systems. Additionally, we describe an error measure that can be computed solely from data and complements any inference technique.

From a dynamical point of view, a tipping point presents perhaps the single most significant threat to an ecological system as it can lead to abrupt species extinction on a massive scale. Climate changes leading to parameter drifts can drive various ecological systems towards a tipping point. Discovering natural and engineering mechanisms to mitigate or delay a tipping point is of considerable interest. We investigate the dynamics of tipping point in multilayer ecological networks supported by mutualism and uncover a natural mechanism that can postpone the occurrence of a tipping point: multiplexity. In particular, for a double-layer mutualistic system of pollinators and plants, coupling between the network layers naturally occurs when there is migration of certain pollinator species from one layer to another. Multiplexity emerges as the migrating species establish their presence in the target layer and therefore have a simultaneous presence in both layers. We demonstrate that the new mutualistic links induced by the migrating species with the residence species have some fundamental benefits to the well being of the ecosystem such as delaying the tipping point and facilitating species recovery. The implication is that articulating and implementing control mechanisms to induce multiplexity can be of significant value to sustaining certain types of ecosystems that are or will be in danger of extinction as the result of environmental changes.

The operational stability of electrical power grids is of utmost importance to ensure a reliable supply of energy and prevent damages and blackouts. Conventional control schemes and grid architecture are challenged by the transition to sustainable energy generation as few, large generators with massive rotating turbines in a highly centralized grid are replaced by many distributed, fluctuating sources of varying size, such as solar and wind power. Therefore, identifying robust and cost-efficient grid architecture, as well as weak points to avoid when planning and building power grids, is an ongoing research area. The voltage and frequency dynamics of AC power grids can be modeled as coupled non-linear oscillators on sparse complex networks. In addition to the desired operating state, in which all nodes are synchronized at network frequency, e.g., 50 Hz, there exists a variety of partially synchronized attractor states. Solitary states consist of a large, synchronized cluster and a single oscillator that rotates with a different velocity, i.e., AC frequency. They pose a threat to power grid stability, as they would cause overload damages and can be easily reached through single-node perturbations. Especially vulnerable to such perturbations are dense sprouts, which are degree-1 nodes with distinct topological properties and a well-connected neighbor. Novel solitary states in which the velocity of the dense sprout differs from its natural velocity have recently been discovered in numerical simulations. In this work, we propose a toy model with which we can theoretically explain the presence of the novel solitary states. It can be used to adjust network architecture to prevent their occurrence. In this model, the rest of the synchronized complex network is reduced to its key factor, i.e. the degree of the neighbor. Applying a linearization approach, we obtain an approximate analytical solution close to the full non-linear dynamics. We then derive a self-consistency equation for the velocity of the solitary node. We demonstrate that the toy model resembles highly localized network modes in the linear stability regime around the operating state. The velocity of the dense sprout arises from resonance with this network mode under the constraint of matching the network‘s power flow. We investigate the stability regime of the novel solitary states and its dependence on initial conditions, system parameters, i.e., characteristics of grid components, and topology.

We show that modulating the coupling of oscillators by noise can induce synchronization in networks. We focus specifically on a two-layer multiplex network of identical Kuramoto oscillators, where each layer is a non-locally coupled ring. We show that in such a system modulating the interlayer coupling by noise can play a similar role to increasing the interlayer coupling strength. In particular, noise can counter-intuitively cause the two layers to synchronize with each other. Noise also induces state transitions in a similar way, in some cases causing the layers to completely synchronize within themselves. We discuss how such disturbances could be beneficial to cognitive function and show how 1/f noise can play an especially efficient role in synchronising the system.

Inferring directed links in networks of interacting systems is a problem spanning many disciplines. Systems out of equilibrium represent a special case, where samples are not independent but structured as timeseries. In this context, Recurrent Neural Networks (RNN) have attracted recent attention, due to their ability to learn dynamical systems from sequences. We introduce a method to infer connectivity of a network from the timeseries of its nodes, using a RNN based on Reservoir Computing (RC). We show how modifications of the standard RC architecture enable a reliable computation of the existence of links between nodes. While the method does not require information about the underlying mathematical model, its performance is further improved if the selection of hyperparameters is roughly informed by knowledge about the system. The method is illustrated with examples from different complex systems, ranging from networks of chaotic Lorenz attractors to biological neurons. Using simulations of these systems, we demonstrate its power and limitations under a variety of conditions, such as noise levels, delayed interactions, size of the network and hidden variables. *joint work with Marie Kempkes and Martin E. Garcia