Basin Stability Research Articles

Large-deviations of the basin stability of power grids.

Energy grids play an important role in modern society. In recent years, there was a shift from using few central power sources to using many small power sources, due to efforts to increase the percentage of renewable energies. Therefore, the properties of extremely stable and unstable networks are of interest. In this paper, distributions of the basin stability, a nonlinear measure to quantify the ability of a power grid to recover from perturbations, and its correlations with other measurable quantities, namely, diameter, flow backup capacity, power-sign ratio, universal order parameter, biconnected component, clustering coefficient, two core, and leafs, are studied. The energy grids are modeled by an Erdős-Rényi random graph ensemble and a small-world graph ensemble, where the latter is defined in such a way that it does not exhibit dead ends. Using large-deviation techniques, we reach very improbable power grids that are extremely stable as well as ones that are extremely unstable. The 1/t-algorithm, a variation of Wang-Landau, which does not suffer from error saturation, and additional entropic sampling are used to achieve good precision even for very small probabilities ranging over eight decades.

Chimera states are fragile under random links

We study the dynamics of coupled systems, ranging from maps supporting chaotic attractors to nonlinear differential equations yielding limit cycles, under different coupling classes, connectivity ranges and initial states. Our focus is the robustness of chimera states in the presence of a few time-varying random links, and we demonstrate that chimera states are often destroyed, yielding either spatiotemporal fixed points or spatiotemporal chaos, in the presence of even a single dynamically changing random connection. We also study the global impact of random links by exploring the Basin Stability of the chimera state, and we find that the basin size of the chimera state rapidly falls to zero under increasing fraction of random links. This indicates the extreme fragility of chimera patterns under minimal spatial randomness in many systems, significantly impacting the potential observability of chimera states in naturally occurring scenarios.

Multistability and attraction basins of discrete-time neural networks with nonmonotonic piecewise linear activation functions

This paper is concerned with multistability and attraction basins of discrete-time neural networks with nonmonotonic piecewise linear activation functions. Under some reasonable conditions, the addressed networks have (2m+1)n equilibrium points. (m+1)n of which are locally asymptotically stable, and the others are unstable. The attraction basins of the locally asymptotically stable equilibrium points are given in the form of hyperspherical regions. These results here, which include existence, uniqueness, locally asymptotical stability, instability and attraction basins of the multiple equilibrium points, generalize and improve the earlier publications. Finally, an illustrative example with numerical simulation is given to show the feasibility and the effectiveness of the theoretical results. The theoretical results and illustrative example indicate that the activation functions improve the storage capacity of neural networks significantly.

Multistability and basin stability in coupled pendulum clocks.

In this paper, we investigate the phenomenon of multistability and the concept of basin stability in two coupled pendula with escapement mechanisms, suspended on horizontally oscillating beam. The dynamics of a single pendulum clock is studied and described, showing possible responses of the unit. The basin stability maps are discussed in two-parameters plane, where we vary both the system's stiffness as well as the damping. The possible attractors for the investigated clocks are discussed, showing that different patterns of synchronization and desynchronization can occur. The oscillators may completely synchronize in one of the three possible combinations (including inphase and antiphase ones), practically synchronize with some fluctuations or stay in the irregular pattern, which includes chaotic motion. The transitions between solutions are studied, uncovering that the road from one type of dynamics into another may become very complex. Moreover, we examine the multistability property of our model using the bifurcation diagrams and the basins of attraction maps, discussing possible scenarios in which the states co-exist. The analysis of attractors' basins uncovers complicated structure of the latter ones, exhibiting that the final behavior of investigated model may be hard to determine and trace. Our results are discussed for the cases of identical and nonidentical pendula, as well as light and heavy beam, showing that depending on considered scenario, various patterns of behaviors and transitions may be observed. The research described in this paper proves that the mechanical properties of the system's suspension may play a crucial role in the possibility of the appearance of different types of attractors and that the basin stabilities of states strictly depend on the values of considered parameters.

On structural and dynamical factors determining the integrated basin instability of power-grid nodes.

In electric power systems delivering alternating current, it is essential to maintain its synchrony of the phase with the rated frequency. The synchronization stability that quantifies how well the power-grid system recovers its synchrony against perturbation depends on various factors. As an intrinsic factor that we can design and control, the transmission capacity of the power grid affects the synchronization stability. Therefore, the transition pattern of the synchronization stability with the different levels of transmission capacity against external perturbation provides the stereoscopic perspective to understand the synchronization behavior of power grids. In this study, we extensively investigate the factors affecting the synchronization stability transition by using the concept of basin stability as a function of the transmission capacity. For a systematic approach, we introduce the integrated basin instability, which literally adds up the instability values as the transmission capacity increases. We first take simple 5-node motifs as a case study of building blocks of power grids, and a more realistic IEEE 24-bus model to highlight the complexity of decisive factors. We find that both structural properties such as gate keepers in network topology and dynamical properties such as large power input/output at nodes cause synchronization instability. The results suggest that evenly distributed power generation and avoidance of bottlenecks can improve the overall synchronization stability of power-grid systems.

Stochastic basins of attraction and generalized committor functions.

We study two generalizations of the basin of attraction of a stable state, to the case of stochastic dynamics, arbitrary regions, and finite-time horizons. This is done by introducing generalized committor functions and studying soujourn times. We show that the volume of the generalized basin, the basin stability, can be efficiently estimated using Monte Carlo-like techniques, making this concept amenable to the study of high-dimension stochastic systems. Finally, we illustrate in a set of examples that stochastic basins efficiently capture the realm of attraction of metastable sets, which parts of phase space go into long transients in deterministic systems, that they allow us to deal with numerical noise, and can detect the collapse of metastability in high-dimensional systems. We discuss two far-reaching generalizations of the basin of attraction of an attractor. The basin of attraction of an attractor are those states that eventually will get to the attractor. In a generic stochastic system, all regions will be left again; no attraction is permanent. To obtain the equivalent of the basin of attraction of a region we need to generalize the notion to cover finite-time horizons and finite regions. We do so by considering soujourn times, the fraction of time that a trajectory spends in a set, and by generalizing committor functions which arise in the study of hitting probabilities. In a simplified setting we show that these two notions reduce to the normal notions of the basin of attraction in the appropriate limits. We also show that the volume of these stochastic basins can be efficiently estimated for high-dimensional systems at computational cost comparable to that for deterministic systems. To fully illustrate the properties captured by the stochastic basins, we show a set of examples ranging from simple conceptual models to high-dimensional inhomogeneous oscillator chains. These show that stochastic basins efficiently capture metastable attraction, the presence of long transients, that they allow us to deal with numerical and approximation noise, and can detect the collapse of metastability with increasing noise in high-dimensional systems.

Finding NHIM: Identifying high dimensional phase space structures in reaction dynamics using Lagrangian descriptors

Phase space structures such as dividing surfaces, normally hyperbolic invariant manifolds, and their stable and unstable manifolds have been an integral part of computing quantitative results such as the transition fraction, rate of stability basin erosion in multi-stable mechanical systems, and rate constants in chemical reactions. Thus, methods that can reveal the geometry of these invariant manifolds in high dimensional phase space (4 or more dimensions) need to be benchmarked by comparing with known results. In this study we assess the capability of one such method called Lagrangian descriptor for revealing the types of high dimensional phase space structures associated with an index-1 saddle in Hamiltonian systems. The Lagrangian descriptor based approach is applied to two and three degree-of-freedom quadratic Hamiltonian systems where the high dimensional phase space structures are known, that is as closed-form analytical expressions. This leads to a direct comparison of features in the Lagrangian descriptor contour maps and the phase space structures’ intersection with an isoenergetic two-dimensional surface, and hence provides a verification of the method.

Noise-induced vegetation transitions in the Grazing Ecosystem

• The analytical expression of the escape probability for the vegetation in the Grazing Ecosystem is derived. • The analytical results are verified with numerical simulations. • The size of the stochastic basin of attraction are explored based on the escape probability. • The effect of herbivore density and stochastic parameter on the basin stability is proposed in the Grazing Ecosystem. The Grazing Ecosystem is a special case of predator-prey systems, which has attracted widespread attention since a long time ago. Because of the ubiquity of noise, there is a growing need to research the influences of noise on the Grazing Ecosystem. This paper is devoted to investigating the transition behaviors of the high vegetation biomass in the Grazing Ecosystem subjected to Gaussian noise and Lévy noise, respectively. Firstly, the original system is translated into the Itô stochastic differential equation, which is utilized to derive the analytical expression of the escape probability through the Dirichlet boundary value problems. Then the transitions between the two vegetation potential wells are explored by calculating the size of the stochastic basin of attraction based on the escape probability. The comparison between the analytical results and the ones through Monte Carlo simulations shows that the proposed method works very well. It turns out that the Gaussian white noise intensity, Lévy noise stability parameter and herbivore density have different impact mechanisms on the basin stability of high density vegetation in the stochastic Grazing Ecosystem.

Lévy noise induced escape in the Morris–Lecar model

The phenomenon of an excitable system producing a pulse under external or internal stimulation may be interpreted as a stochastic escape problem. This work addresses this issue by examining the Morris–Lecar neural model driven by symmetric α-stable Lévy motion (non-Gaussian noise) as well as Brownian motion (Gaussian noise). Two deterministic quantities: the first escape probability and the mean first exit time, are adopted to analyze the state transition from the resting state to the excited state and the stability of this stochastic model. Additionally, a recent geometric concept, the stochastic basin of attraction is used to explore the basin stability of the escape region. Our main results include: (i) the larger Lévy motion index with smaller jump magnitude and the relatively small noise intensity are conducive for the Morris–Lecar model to produce pulses; (ii) a smaller noise intensity and a larger Lévy motion index make the mean first exit time longer, which means the stability of the resting state can be enhanced in this case; (iii) the effect of ion channel noise is more pronounced on the stochastic Morris–Lecar model than the current noise. This work provides some numerical simulations about the impact of non-Gaussian, heavy-tailed, burst-like fluctuations on excitable systems such as the Morris–Lecar system.

Order parameter allows classification of planar graphs based on balanced fixed points in the Kuramoto model.

Phase balanced states are a highly underexplored class of solutions of the Kuramoto model and other coupled oscillator models on networks. So far, coupled oscillator research focused on phase synchronized solutions. Yet, global constraints on oscillators may forbid synchronized state, rendering phase balanced states as the relevant stable state. If, for example, oscillators are driving the contractions of a fluid filled volume, conservation of fluid volume constrains oscillators to balanced states as characterized by a vanishing Kuramoto order parameter. It has previously been shown that stable, balanced patterns in the Kuramoto model exist on circulant graphs. However, which noncirculant graphs first of all allow for balanced states and what characterizes the balanced states is unknown. Here, we derive rules of how to build noncirculant, planar graphs allowing for balanced states from the simple cycle graph by adding loops or edges to it. We thereby identify different classes of small planar networks allowing for balanced states. Investigating the balanced states' characteristics, we find that the variance in basin stability scales linearly with the size of the graph for these networks. We introduce the balancing ratio as an order parameter based on the basin stability approach to classify balanced states on networks and evaluate it analytically for a subset of the network classes. Our results offer an analytical description of noncirculant graphs supporting stable, balanced states and may thereby help to understand the topological requirements on oscillator networks under global constraints.

Early warning and basin stability in a stochastic vegetation-water dynamical system

The dynamical evolution of a vegetation-water system subjected to Gaussian noise disturbance is explored. With regard to the regime shift of the stationary response of the system, the early warning signal can be detected by the maximum of the stationary probability density function. And for the transient response of the bistable system, the basin stability is investigated by two different deterministic quantities, the first escape probability and the mean first exit time. Firstly, the stochastic system with two variable is translated into the Itô stochastic differential equation to derive the analytical expression of the first escape probability and the mean first exit time with different Balayage-Dirichlet exterior boundary condition. Then, the two indexes are calculated by the finite difference method, respectively. The results, compared with the ones through Monte Carlo simulations, shows that the proposed method works very well. It turns out that the increase of noise intensity induces advance of the early warning and decreases basin stability of the vegetation-water system.

Spike chimera states and firing regularities in neuronal hypernetworks.

A complex spatiotemporal pattern with coexisting coherent and incoherent domains in a network of identically coupled oscillators is known as a chimera state. Here, we report the emergence and existence of a novel type of nonstationary chimera pattern in a network of identically coupled Hindmarsh-Rose neuronal oscillators in the presence of synaptic couplings. The development of brain function is mainly dependent on the interneuronal communications via bidirectional electrical gap junctions and unidirectional chemical synapses. In our study, we first consider a network of nonlocally coupled neurons where the interactions occur through chemical synapses. We uncover a new type of spatiotemporal pattern, which we call "spike chimera" induced by the desynchronized spikes of the coupled neurons with the coherent quiescent state. Thereafter, imperfect traveling chimera states emerge in a neuronal hypernetwork (which is characterized by the simultaneous presence of electrical and chemical synapses). Using suitable characterizations, such as local order parameter, strength of incoherence, and velocity profile, the existence of several dynamical states together with chimera states is identified in a wide range of parameter space. We also investigate the robustness of these nonstationary chimera states together with incoherent, coherent, and resting states with respect to initial conditions by using the basin stability measurement. Finally, we extend our study for the effect of firing regularity in the observed states. Interestingly, we find that the coherent motion of the neuronal network promotes the entire system to regular firing.

Global phase space structures in a model of passive descent

Even the most simplified models of falling and gliding bodies exhibit rich nonlinear dynamical behavior. Taking a global view of the dynamics of one such model, we find an attracting invariant manifold that acts as the dominant organizing feature of trajectories in velocity space. This attracting manifold captures the final, slowly changing phase of every passive descent, providing a higher-dimensional analogue to the concept of terminal velocity, the terminal velocity manifold. Within the terminal velocity manifold in extended phase space, there is an equilibrium submanifold with equilibria of alternating stability type, with different stability basins. In this work, we present theoretical and numerical methods for approximating the terminal velocity manifold and discuss ways to approximate falling and gliding motion in terms of these underlying phase space structures.

Power and Stability Policies in the Black Sea-Caspian Region in the Context of Max Weber’s Authority Approach

Weber defined the authority that gave legitimacy to power based on the concept of sovereignty and tried to understand the level of awareness of the individual’s social actions. Together with the phenomenon of power, who played an active role in the strategic platform formed at the same time the hegemony of the authority. The playmakers, who are the active subjects of International Relations, produce proactive-structural policies considering the power balances. In this context, the Black Sea and the Caspian Basin are not only affected by energy, politics and power policies on geopolitical axes but also affect these policies. Turkey is strengthening its central position taking an active role in the Black Sea and the Caspian Basin energy and stability policies for the balance of power in the Eurasian axis. In this study, on the scale of the Black Sea and the Caspian Basin; in terms of Turkey, regional and global actors in energy policies, geopolitics and the balance of power will be examined.

Dynamic environment-induced multistability and critical transition in a metacommunity ecosystem.

We study a metacommunity model of consumer-resource populations coupled via dispersal under an environment-dependent framework, and we explore the occurrence of multistability and critical transition. By emphasizing two magnitudes acting on a dynamic environment at temporal and spatial scales, the coupled system with simple diffusive coupling and the nonlinear environmental coupling enables various interesting complex dynamics such as bistability, multistability, and critical transitions. Using the basin stability measure, we find the probability of attaining each alternative state in a multistable region. In addition, critical transitions (one from a high to a low species density and the other from a low to a high species density) are identified at different magnitudes in the presence of stochastic fluctuations. We also explore the robustness of critical slowing-down indicators, e.g., lag-1 autocorrelation and variance, to forewarn the critical transition in the metacommunity model. Further, a network structure also identifies synchronization and multiclustering for a different choice of initial conditions. In contrast with the earlier studies on dynamic environmental coupling, our results based on the defined magnitudes provide important insights into environmental heterogeneity, which determines the set of environmental conditions to predict metacommunity stability and persistence.

A Basin Stability Based Metric for Ranking the Transient Stability of Generators

This paper discusses a simulation-based generator-level stability index and demonstrates the use of the index to quantify the impact of different load scenarios, transmission line reactances, and generator parameters on transient stability. A basin stability concept that is based on recent works in the physics community on complex systems and power systems is used, which estimates the regions of attraction through time-domain simulation. The quantitative results generated by basin stability can provide a general ranking metric to reflect which generator or group of generators is likely to become unstable after disturbances. A detailed subtransient model with sixth-order subtransient generators, second-order exciters, and third-order power system stabilizers has been used for discussion. Additionally, a multinode basin stability study on a 16-machine 68-bus system with the swing-equation model is also included to demonstrate the replicability. The state-of-the-art parallel computing techniques can significantly reduce the simulation time of the proposed method.

Control of hierarchical networks by coupling to an external chaotic system

We explore the behaviour of chaotic oscillators in hierarchical networks coupled to an external chaotic system whose intrinsic dynamics is dissimilar to all the oscillators in the network. We find that coupling to one such dissimilar external system manages to suppress the chaotic dynamics of all the oscillators at all levels of the network, at sufficiently high coupling strength. The chaos suppression is independent of the system size and occurs irrespective of whether the connection to the external system is direct, or indirect through oscillators at another level in the hierarchy. Though the steady states vary across different tiers, the oscillators are synchronized to the same steady state at a particular level of hierarchy. Lastly, we quantify the efficacy of control by estimating a global stability measure analogous to the basin stability of the emergent steady state. Our quantitative results explicitly indicate the easy controllability of hierarchical networks of chaotic oscillators by one dissimilar chaotic system, thereby suggesting a potent method that may help design control strategies.

Investigation of the potential of buffalo and couch grasses to grow on AFIs and for removal of nutrients from paper mill wastewater.

The potential growth of buffalo grass (Stenotaphrum secundatum) and couch grass (Cynodon dactylon) on artificial floating islands (AFIs) and their ability to remove total nitrogen (TN) and total phosphorus (TP) from a simulated paper mill wastewater was studied. This was done to assess the potential of AFIs for removal of nutrients from aerated stabilization basins (ASBs) that had occasional growth of blue-green algae (BGA) to bloom levels. Small scale AFIs were prepared using polyethylene foam and planted with the grasses in 30 L of tested water. Trials were conducted in a plastic covered greenhouse over a three-month period where temperatures ranged from 15 to 44 °C. The results showed that both buffalo and couch grasses can adapt to planting in AFIs showing increases of 125% and 148% in wet weight, respectively. Nutrient uptake by buffalo grass and couch grass were found to be similar. Percentage uptakes of TP and TN from the synthetic water by the buffalo grass were 82% and 47%, whereas by couch grass, uptakes were 83% and 45%, respectively.

Synchronizability of two neurons with switching in the coupling

Time-varying networks are prominently present in the nervous system, where the connections between neurons vary according to the activity of pre and post synapses. Using this as motivation, we here consider a system of two ephaptically coupled neurons with a periodically time-varying link between the two membrane potentials. We derive the master stability function in dependence on the membrane potential coupling strength and in dependence on the ephaptic coupling strength. We first determine coupling ranges at which the two neurons are synchronizable if the link between them is static. In doing so, we find that the ephaptic coupling has a negligible impact on synchronization, while the coupling through the membrane potential has a strong effect. We then consider switching in the coupling, determining the average synchronization error for a fixed ephaptic coupling strength and different membrane potential coupling strengths, and for various switching frequencies. We find the values of the switching coefficient that can synchronize the two neurons for each of the considered switching frequencies, and finally, we also use the basin stability method to determine the global stability of synchronization.

Emergence of synchronization in multiplex networks of mobile Rössler oscillators.

Different aspects of synchronization emerging in networks of coupled oscillators have been examined prominently in the last decades. Nevertheless, little attention has been paid on the emergence of this imperative collective phenomenon in networks displaying temporal changes in the connectivity patterns. However, there are numerous practical examples where interactions are present only at certain points of time owing to physical proximity. In this work, we concentrate on exploring the emergence of interlayer and intralayer synchronization states in a multiplex dynamical network comprising of layers having mobile nodes performing two-dimensional lattice random walk. We thoroughly illustrate the impacts of the network parameters, in particular, the vision range ϕ and the step size u together with the inter- and intralayer coupling strengths ε and k on these synchronous states arising in coupled Rössler systems. The presented numerical results are very well validated by analytically derived necessary conditions for the emergence and stability of the synchronous states. Furthermore, the robustness of the states of synchrony is studied under both structural and dynamical perturbations. We find interesting results on interlayer synchronization for a continuous removal of the interlayer links as well as for progressively created static nodes. We demonstrate that the mobility parameters responsible for intralayer movement of the nodes can retrieve interlayer synchrony under such structural perturbations. For further analysis of survivability of interlayer synchrony against dynamical perturbations, we proceed through the investigation of single-node basin stability, where again the intralayer mobility properties have noticeable impacts. We also discuss the scenarios related mainly to effects of the mobility parameters in cases of varying lattice size and percolation of the whole network.