Basin Stability Research Articles

State-Space Basins for Monopedal Jumping with Stable Landing

Abstract Maintaining stability during jumping remains a challenge due to its hybrid dynamics. Despite recent advancements in the control of jumping robots, existing research lacks an accurate and comprehensive criterion for the stability of jumping. This study proposes a general criterion for jumping that incorporates the system’s nonlinear dynamics and environmental constraints. The stability basins are formulated as a state-space partition based on a rigorous definition extending the balanced state basin (BSB) concept, which has been validated as a stability criterion for biped robots in legged stance. A hybrid approach, using a combination of analytical and optimization methods, solves the flight and stance phases of jumping as independent sub-problems. The basins are computed for a monopedal jumping robot, Salto-1P, and validated with simulations. To demonstrate its general applicability, two distinct flight-to-stance tasks are considered: targeted jumping and cat-like righting. Analysis of each task’s stability basins and the landing BSB reveals the inherent trade-offs between stability and the task requirements for variations in the initial state variables such as speed, orientation, and height. The stability basins represent the system’s fundamental controller-independent stability characteristics and can be applied to the mechanical design, control, and fall prediction of jumping robots.

Vibration-resonance chimeras in coupled excitable systems with heterogeneous aperiodic excitations

In this work, we present a novel chimera-like state known as the vibration-resonance chimera (VRC) state, which is induced by heterogeneous aperiodic (HA) excitations and emerges within the optimal range of amplitude for the HA excitations. We investigate the dynamic behaviors of the FitzHugh-Nagumo model in relation to HA excitations and coupling strength in parameter spaces, identifying specific regions where VRC states exist and analyzing the transition mechanisms between different dynamic behaviors. Using basin stability measures, we find that the mean period of HA excitations affects the multi-stable states of neural systems. Specifically, the model excited by HA excitations with a larger mean period is more likely to exhibit multi-stable states. Additionally, we explore the effects of two levels of heterogeneity in HA excitations, namely amplitude heterogeneity and period heterogeneity, on VRC states. Amplitude heterogeneity can suppress the occurrence of VRC states, whereas period heterogeneity can promote their generation. This work provides valuable insights into complex system dynamics, enhancing our understanding of chimera states in neuronal models and their potential applications in neural networks.

Deeper but smaller: Higher-order interactions increase linear stability but shrink basins.

A key challenge of nonlinear dynamics and network science is to understand how higher-order interactions influence collective dynamics. Although many studies have approached this question through linear stability analysis, less is known about how higher-order interactions shape the global organization of different states. Here, we shed light on this issue by analyzing the rich patterns supported by identical Kuramoto oscillators on hypergraphs. We show that higher-order interactions can have opposite effects on linear stability and basin stability: They stabilize twisted states (including full synchrony) by improving their linear stability, but also make them hard to find by markedly reducing their basin size. Our results highlight the importance of understanding higher-order interactions from both local and global perspectives.

Basin stability for updating system uncertainties.

In this paper, we propose an application of the basin stability tool which allows us to update the information on the system properties under parameter uncertainties. The concept is presented using a classical mechanical setup of coupled pendula, exchanging the energy via the supporting structure. Depending on the support parameters, the model can exhibit different types of coexisting synchronous patterns as well as remaining desynchronized. We calculate basin stability maps of particular behaviors and combine them with prior parameter distributions using Bayesian inference. The obtained posterior distributions, based on the attractor occurrence, update our knowledge on the system properties in the terms of probabilities. We also underline the problem of evaluating basin stability close to the existence borders, comparing the classical approach of fixed parameters with the one involving variations. The differences between the estimation methods can have a crucial meaning for the discussed application and should be considered carefully. The results presented in this paper uncover ways of applying the basin stability concept, which can be used to study the properties of complex dynamical systems from a probability perspective.

Tipping points, multistability, and stochasticity in a two-dimensional traffic network dynamics

Mitigating traffic jams is a critical step for the betterment of the urban transportation system, which comprises a large number of interconnected routes to form an intricate network. To understand distinct features of vehicular traffic flow on a network, a macroscopic two-dimensional traffic network model is proposed incorporating intra-nodal and inter-nodal vehicular interaction. Utilizing the popular techniques of nonlinear dynamics, we investigate the impact of different parameters like occupancy, entry rates, and exit rates of vehicles. The existence of saddle-node, Hopf, homoclinic, Bogdanov–Takens, and cusp bifurcations have been shown using single or biparametric bifurcation diagrams. The occurrences of different multistability (bistability/tristability) phenomena, stochastic switching, and critical transitions are explored in detail. Further, we calculate the possibility of achieving each alternative state using the basin stability metric to characterize multistability. In addition, critical transitions from free flow to congestion are identified at different magnitudes of stochastic fluctuations. The applicability of critical slowing down based generic indicators, e.g., variance, lag-1 autocorrelation, skewness, kurtosis, and conditional heteroskedasticity are investigated to forewarn the critical transition from free flow to traffic congestion. It is demonstrated through the use of simulated data that not all of the measures exhibit sensitivity to rapid phase transitions in traffic flow. Our study reveals that traffic congestion emerges because of either bifurcation or stochasticity. The result provided in this study may serve as a paradigm to understand the qualitative behavior of traffic jams and to explore the tipping mechanisms occurring in transport phenomena.

Mass variation effect in CR3B problem with nanoscale

The aim of this paper is to investigate the dynamical behavior of a Carbon atom under the forces of the two fullerenes in the circular restricted three-body (CR3B) configuration with nanoscale. Here, we assume that the mass of a Carbon atom varies according to the Jeans law, and then with the use of Meshcherskii transformation, we determine the equations of motion and quasi-Jacobian integral. Then by the use of numerical values of the parameters, we investigate the potential surfaces, equilibrium points, regions of motion, Poincaré surfaces of sections, periodic orbits, basins of attraction and stability of the equilibrium points.

A Unified Approach for the Calculation of Different Sample-Based Measures with the Single Sampling Method

This paper explores two sample-based methods for analysing multistable systems: basin stability and basin entropy. Both methods rely on many numerical integration trials conducted with diverse initial conditions. The collected data is categorised and used to compute metrics that characterise solution stability, phase space structure, and system dynamics predictability. Basin stability assesses the overall likelihood of reaching specific solutions, while the basin entropy measure aims to capture the structure of attraction basins and the complexity of their boundaries. Although these two metrics complement each other effectively, their original procedures for computation differ significantly. This paper introduces a universal approach and algorithm for calculating basin stability and entropy measures. The suitability of these procedures is demonstrated through the analysis of two non-linear systems.

Developing climate change adaptation pathways in the agricultural sector based on robust decision-making approach (case study: Sefidroud Irrigation Network, Iran).

Allocation of water in the situation of climate change presents various uncertainties. Consequently, decisions must be made to ensure stability and functionality across different climatic scenarios. This study aims to examine the effectiveness of adaptation strategies in the agricultural sector, including a 5% increase in irrigation efficiency (S1) and a shift in irrigation method to Dry-DSR (direct seeded rice) under conditions of climatic uncertainty using a decision-making approach. The study focuses on the basin downstream of the Sefidroud dam, encompassing the Sefidroud irrigation and drainage network. Initially, basin modeling was conducted using the WEAP integrated management software for the period 2006-2020. Subsequently, the impact of climate change was assessed, considering RCP2.6, RCP4.5, and RCP8.5 emission scenarios on surface water resources from 2021 to 2050. Runoff and cultivated area, both subject to uncertainty, were identified as key parameters. To evaluate strategy performance under different uncertainties and determine the efficacy of each strategy, regret and satisfaction approaches were employed. Results indicate a projected decrease in future rainfall by 3.5-11.8% compared to the base period, accompanied by an increase in maximum and minimum temperatures (0.83-1.62 °C and 1.15-1.33 °C, respectively). Inflow to the Sefidroud dam is expected to decrease by 13-28%. Presently, the Sefidroud irrigation and drainage network faces an annual deficit of 505.4 MCM, and if current trends persist with the impact of climate change, this shortfall may increase to 932.7 MCM annually. Furthermore, satisfaction indices for strategy (S2) are 0.77 in an optimistic scenario and 0.70 in strategy (S1). In a pessimistic scenario, these indices are 0.67 and 0.56, respectively. Notably, changing the irrigation method with Dry-DSR is recommended as a robust strategy, demonstrating the ability to maintain basin stability under a broad range of uncertainties and climate change scenarios. It is crucial to note that the results solely highlight the effects of climate change on water sources entering the Sefidroud dam. Considering anthropogenic activities upstream of the Sefidroud basin, water resource shortages are expected to increase. Therefore, reallocating water resources and implementing practical and appropriate measures in this area are imperative.

Incommensurate fractional-order analysis of a chaotic system based on interaction between dark matter and dark energy with engineering applications

Chaotic systems, characterized by their sensitivity to initial conditions and the presence of deterministic unpredictability, have garnered significant attention in various fields of science and engineering. Researchers have demonstrated a growing interest in recent years in identifying various physical systems as chaotic system, including but not limited to fluid dynamics, electrical circuits, and biological oscillators. In this study, a cosmological chaotic system that portrays the interaction between dark matter and dark energy is examined using incommensurate fractional-order analysis. The effects of incommensurate fractional orders are evaluated by means of phase portraits, bifurcation diagrams and Lyapunov exponents spectra. It is dedicated that wider chaotic regions are observed once the values of these incommensurate fractional orders are changed. Meanwhile, a hidden attractor is found based on the stability analysis and basin of attraction in the incommensurate fractional-order dark matter and dark energy (IFODMDE) chaotic system. These findings highlight the richer dynamic characteristics of the IFODMDE chaotic system. Furthermore, the identified hidden attractor is effectively employed in this study for various engineering applications, including chaos control, random number generator (RNG) design, and image encryption. The analytical results offer a more accurate description of real-world physical phenomena, thanks to the inclusion of incommensurate fractional orders. The application results prove the good randomness of the IFODMDE chaotic system.

How do predator interference, prey herding and their possible retaliation affect prey-predator coexistence?

<abstract><p>In this paper, focusing on individualistic generalist predators and prey living in herds which coexist in a common area, we propose a generalization of a previous model, namely, a two-population system that accounts for the prey response to predator attacks. In particular, we suggest a new prey-predator interaction term with a denominator of the Beddington-DeAngelis form and a function in the numerator that behaves as $ N $ for small values of $ N $, and as $ N^{\alpha} $ for large values of $ N $, where $ N $ denotes the number of prey. We can take the savanna biome as a reference example, concentrating on large herbivores inhabiting it and some predators that feed on them. Only two conditionally stable equilibrium points have emerged from the model analysis: the predator-only equilibrium and the coexistence one. Transcritical bifurcations from the former to the latter type of equilibrium, as well as saddle-node bifurcations of the coexistence equilibrium have been identified numerically by using MATLAB. In addition, the model was found to exhibit bistability. Bistability is studied by using the MATLAB toolbox bSTAB, paying particular attention to the basin stability values. Comparison of coexistence equilibria with other prey-predator models in the literature essentially shows that, in this case, prey thrive in greater numbers and predators in smaller numbers. The population changes due to parameter variations were found to be significantly less pronounced.</p></abstract>

Long-term impacts of urban floodplain management and habitat restoration on lizard communities in a Sonoran Desert city

Riparian zones are critical for biodiversity but have suffered widespread degradation and therefore are key targets for ecological restoration. In urban contexts, restoring riparian ecosystems can have widespread benefits but these efforts are often limited by needs to mitigate flood risks and by a lack of long-term monitoring to evaluate efficacy, presenting important challenges for managers. To evaluate restoration potential in urban riparian systems, we studied responses of a diverse Sonoran Desert lizard community to combined flood control and habitat restoration efforts along a major river course and tributary drainage over 15 years in urban Tucson, Arizona. We used a before/after-control/impact design and linear mixed-effect models to estimate species responses to treatments that included bank stabilization and detention basin construction for flood and erosion control combined with passive water harvesting, native vegetation planting and seeding, and targeted creation of microhabitats. Despite marked declines of relative abundances and richness at treated sites immediately following construction, overall impacts were mostly positive or neutral, and negative for just one of six focal species. Most species recovered to pre-treatment levels of abundance within 2–3 years, but recovery dynamics varied with differences in life history traits and habitat use among species. In general, widely-moving, faster-maturing terrestrial species recovered more quickly and responded most positively to restoration, whereas more arboreal species with longer generation times and smaller home ranges took longer to benefit from restoration treatments. Few studies assess impacts of paired urban flood control and habitat restoration on wildlife, and our findings suggest management of urban riparian systems can provide simultaneous benefits to humans and wildlife. Future efforts may be improved by preserving and fostering more heterogeneous cover including key resources such as mature trees, as well as potentially translocating animals into restored but disconnected patches of habitat in urbanized landscapes.

Basin delimitation and stability assessment for power systems in the projective space

This paper presents a method for determining the global basin boundary of the power systems. To show the general outline of the basin, the power system is shrink to the projective space. First, a new vector field is induced according to Poincare’s method. After the transformation, the structural stability of the system remains the same. The singularities at infinity are obtained by the proposed shrinking-projection method. The boundary of the basin which is extended to the infinity are complemented by the singularities at infinity in the projective space. The stability of the power system is then evaluated based on the distribution of the singularities at infinity. Finally, the proposed method is applied to the power system with perturbations to study the characteristic of the stochastic basin boundary. The validity of the proposed method is demonstrated on a 9-bus, 11-bus and 68-bus test system.

ENSO may have contributed to sea level changes in the Gulf of Thailand during the Late-Holocene

The Gulf of Thailand is ideal for studying eustatic sea level fluctuations in Southeast Asia due to its shallow basin and tectonic stability. However, our understanding of how this region’s relative sea level (RSL) has fluctuated over the Holocene epoch is far from complete. In this study, we used lithostratigraphy, loss on ignition, grain size, and pollen analyses to reconstruct the environmental changes in the Sam Roi Yot wetland, which was significantly influenced by seawater intrusion, driven by fluctuations in RSL in the Gulf of Thailand. Therefore, the analyzed pollen records of the sediment core from the wetland reflected variabilities in the RSL in the Gulf of Thailand. Subsequently, we found that after a sea level highstand prior to 4000 cal y BP, the RSL gradually fell with two significant regressions at c. 2950 and 1850–1450 cal y BP before rising at 1450–1050 cal y BP and declining after that. The inconsistency between RSL reconstruction based on our results and the global sea level changes simulated by the Glacial Isostatic Adjustment (GIA) model further suggests that long–term El Niño Southern Oscillations (ENSO) variabilities may have played a significant role in sea level changes in the Gulf of Thailand over the Late-Holocene period. Thus, during extended El Niño or La Niña conditions, the sea level would have been consistently lower or higher than expected from eustatic and isostatic processes alone. Overall, this study emphasizes the importance of considering regional factors such as ENSO to understand sea level changes in Southeast Asia.

Cross-correlated sine-Wiener noises-induced transitions in a tumor growth system

Cross-correlated sine-Wiener (CCSW) noises-induced transitions in a stochastic tumor growth dynamical system are studied. Utilizing Fox’s approach and Hänggi ansatz, the role of CCSW noises on dynamical systems is equivalently transformed into that of multiplicative Gaussian white noise, effectively solving the dynamics puzzles caused by CCSW noises and facilitating the intractable theoretical investigation related to CCSW noises. The impacts of CCSW noises on transitions between the high-population (undesirable) basin B1 and low-population (desirable) basin B2, available to describe the process of tumor elimination and recurrence, are examined theoretically and numerically from the perspective of basin stability through the first escape probability (FEP) and mean first exit time (MFET). Lower FEP or longer MFET characterizing stronger stability reflects a lower likelihood of transitions. Four stability indexes, δE, δSBA, δR, and δSBA∗, are considered to fully explore the role of CCSW noises on transitions between B1 and B2 to reveal its contribution during different tumor treatment stages. We discover that (i) the CCSW noise intensities exert a dual effect on transitions: for λ≤0 (cross-correlation strength), enhanced noise intensities lead to smaller δE (/δSBA) or larger δR (or smaller δSBA∗), acting to weaken the undesirable or desirable basin stability and favor transitions; for λ>0, critical noise intensities that maximize δE (/δSBA) or minimize δR (or maximize δSBA∗) exist, indicating that noise-enhanced stability occurs, which can be reinforced by enhancing λ. (ii) Increasing the correlation time τ leads to weaker stability and higher transition probability, while λ acts oppositely in inducing transitions. Therefore, to achieve better tumor eradication efficacy, destabilizing the undesirable basin B1 by enhancing the negatively correlated SW noise intensities and increasing τ is urgent. During the stage of suppressing tumor recurrence, the unsafe parameter domain that causes a larger δR, implying a high risk of re-entry into B1, should be eschewed. Instead, the positively correlated critical noise intensities with a larger λ and smaller τ for stabilizing the desirable basin B2 are preferable.

Dynamics of a self-propelled capsule robot in contact with different folds in the small intestine

Considering the anatomy of small intestine involving lesions, circular folds and tumours are the major sources resisting the locomotion of capsule robots. By mimicking the small-bowel tumours as cone folds, this paper presents a comparative study on the dynamics of a vibro-impact capsule robot in contact with different circular and cone folds. With the aid of GPU parallel computing and path-following techniques, extensive bifurcation and basin stability analyses are performed to identify different capsule-fold interactions and unravel the parametric influences on the robot, such as fold shape, Young’s modulus and robot’s control parameters (e.g., excitation period and amplitude). It is found that fold shape and Young’s modulus may only affect capsule’s dynamics significantly when robot’s excitation period is large. Two essential locomotion modes, a period-one motion with capsule-fold contact in the small region of excitation amplitude and a fold crossing motion in the large region of excitation amplitude, dominating the dynamics of the robot regardless of fold shape and Young’s modulus are observed. In addition, the instability mechanism of this period-one motion is revealed in detail. The numerical study presented in this work will provide a solid basis for the locomotion control of the robot when encountering different types of circular folds and small-bowel tumours. It also offers the potential of utilising robot’s dynamics for bowel cancer detection.

Tumor state transitions driven by Gaussian and non-Gaussian noises

Tumor state transitions between the excited (high-concentration) and nonexcited (low-concentration) basins under the Gaussian white noise and non-Gaussian colored noise are investigated via the most probable steady states (MPSS) and the first escape probability (FEP)-based stochastic basin of attraction (SBA), respectively. Reducing the non-Gaussian colored noise and then utilizing the unified colored noise approximation (UCNA), the Markov system is derived. The extremal controlling equation of stationary probability density function (SPDF) is derived to analyze the impacts of noise on transitions in terms of MPSS. The existence of the ‘color’ of the non-Gaussian colored noise induces the reappearance of the uncorrelated additive white noise parameter that had vanished from the extremal controlling equation, completely reversing the inability of the uncorrelated additive Gaussian white noise to operate on transitions. The FEP-dependent SBA characterizing the excited basin stability is performed to further analyze the role of noise on the likelihood of escaping to the nonexcited state. Results show that the cross-correlated noises play a dual role in regulating SBA. The increased SBA indicating more difficulty to escape to the nonexcited state reflects a worse therapeutic effect. Therefore, enhancing the negatively correlated noise intensities and augmenting the non-Gaussian noise correlation time is essential for destabilizing the excited basin and achieving optimal therapeutic efficacy.

A complex networks based approach to nonlinear aeroelasticity

The present study investigates the effect of low intensity additive noise on a subcritical pitch-plunge aeroelastic system through the lens of complex networks. In the deterministic system, the 0→ fixed point loses its stability at the flutter point, via subcritical Hopf bifurcation. The unstable LCO becomes a high-amplitude stable LCO at the turning point, posing danger to structural integrity. The system displays bi-stability, and an analysis of the basin stability has revealed it to be highly susceptible to external perturbations. The external perturbations, modelled here as a direct random forcing in the form of a Gaussian white noise, could drastically change the deterministic dynamics. It brought in new dynamical states like noise induced intermittency and an advancement in the onset of LCO. Phenomenological and Dynamical bifurcations (for the stochastic system) were tracked and found to capture only the qualitative changes. To quantify the changes in the dynamics, response time-series were converted into complex networks, based on recurrences in the phase-space. Various network theory based measures such as edge density, clustering coefficient etc., were successful in capturing the dynamical transitions well quantitatively and to potentially serve as precursors for the aeroelastic system. Average path length, global efficiency measures signified the average separation of states and the robustness of the attractor, respectively. The networks also displayed the famous “small-world” property, indicating that transition between states occurs through very few intermediate states. The results have shown that a careful consideration of the stochastic perturbations is needed in the design and operations of such systems, and that complex networks based methods are successful in providing novel perspectives on the topology of the phase-space.

Coupled pendula with varied forcing direction.

In this paper, we investigate the complex dynamics of rotating pendula arranged into a simple mechanical scheme. Three nodes forming the small network are coupled via the horizontally oscillating beam (the global coupling structure) and the springs (the local coupling), which extends the research performed previously for similar models. The pendula rotate in different directions, and depending on the distribution of the latter ones, various types of behaviors of the system can be observed. We determine the regions of the existence and co-existence of particular solutions using both the classical method of bifurcations, as well as a modern sample-based approach based on the concept of basin stability. Various types of states are presented and discussed, including synchronization patterns, coherent dynamics, and irregular motion. We uncover new schemes of solutions, showing that both rotations and oscillations can co-exist for various pendula, arranged within one common system. Our analysis includes the investigations of the basins of attraction of different dynamical patterns, as well as the study on the properties of the observed states, along with the examination of the influence of system's parameters on their behavior. We show that the model can respond in spontaneous ways and uncover unpredicted irregularities occurring for the states. Our study exhibits that the inclusion of the local coupling structure can induce complex, chimeric dynamics of the system, leading to new co-existing patterns for coupled mechanical nodes.

Resilience basins of complex systems: An application to prosumer impacts on power grids.

Comparable to the traditional notion of stability in system dynamics, resilience is typically measured in a way that assesses the quality of a system's response, for example, the speed of its recovery. We present a broadly applicable complementary measurement framework that quantifies resilience similarly to basin stability by estimating a resilience basin, which reflects the extent of adverse influences that the system can recover from in a sufficient manner. In contrast to basin stability, the adverse influences considered here are not necessarily displacements in state space, but arbitrarily complex impacts to the system, quantified by adequate parameters. As a proof of concept, we present two applications: (i) the well-studied single-node power system as an easy-to-follow example and (ii) a stochastic model of a low-voltage DC power grid undergoing an unregulated energy transition consisting in the random appearance of prosumers. These act as decentral suppliers of photovoltaic power and alter the flow patterns while the grid topology remains unchanged. The resilience measurement framework is applied to evaluate the effect and efficiency of two response options: (i) upgrading the capacity of existing power lines and (ii) installing batteries in the prosumer households. The framework demonstrates that line upgrades can provide potentially unlimited resilience against energy decentralization, while household batteries are inherently limited (achieving ≤70% of the resilience of line upgrades). Further, the framework aids in optimizing budget efficiency by pointing toward threshold budget values as well as budget-dependent ideal strategies for the allocation of line upgrades and for the battery charging algorithm.

Lévy noise influences basin stability in a delayed vegetation-water dynamical system.

The stochastic stability for the irregular attraction basin in a time-delayed vegetation-water ecosystem disturbed by Lévy noise is explored. We first discuss that average delay time does not change the attractors of the deterministic model but affects the corresponding attraction basins, and we present the generation of Lévy noise. Then, we investigate the influence of stochastic parameters and delay time on the ecosystem by two statistical indicators, the first escape probability (FEP) and the mean first exit time (MFET). The numerical algorithm for calculating the FEP and the MFET in the irregular attraction basin is implemented, which is effectively verified by Monte Carlo simulations. Furthermore, the metastable basin is defined by the FEP and the MFET and confirms the consistency of the two indicators reflecting results. The result shows that the stochastic stability parameter, especially the noise intensity, decreases the basin stability of the vegetation biomass. In this environment, the time delay effect can validly alleviate its instability.