Different Basins Of Attraction Research Articles

An asymmetric rotor model under external, parametric, and mixed excitations: Nonlinear bifurcation, active control, and rub-impact effect

This article explores the nonlinear dynamical analysis and control of a [Formula: see text] DOF system that emulates the lateral vibration of an asymmetric rotor model under external, multi-parametric, and mixed excitation. The linear integral resonant controller ([Formula: see text]) has been coupled to the rotor as an active damper through a magnetic actuator. The complete mathematical model, governing the nonlinear interaction among the rotor, controller, and actuator, is derived based on electromagnetic theory and the principle of solid mechanics. This results in a discontinuous [Formula: see text] DOF system coupled with two [Formula: see text] DOF systems, incorporating the rub-impact effect between the rotor and stator. The complicated mathematical model is investigated using analytical techniques, employing the perturbation method, and validated numerically through time response, basins of attraction, bifurcation diagrams, [Formula: see text] chaotic test, and Poincaré return map. The main findings indicate that the asymmetric system model may exhibit nonzero bistable forward whirling motion under external excitation. Additionally, it can whirl either forward or backward under multi-parametric excitation, besides the trivial stable solution. Furthermore, in the case of mixed excitation, the rotor displays nontrivial tristable solutions, with two corresponding to forward whirling orbits and the other one corresponding to backward whirling oscillation. These findings are validated through the establishment of different basins of attraction. Finally, the performance of the [Formula: see text] in mitigating rotor vibrations and averting nonlinear catastrophic bifurcations under various excitation conditions. Furthermore, the rotor’s dynamical behavior and stability are explored in the event of an abrupt failure of one of the connected controllers. The outcomes demonstrate that the proposed [Formula: see text] effectively eliminates dangerous nonlinearities, steering the system to respond akin to a linear system with controllable oscillation amplitudes. However, the sudden controller failure induces a local rub-impact effect, leading to a nonlocal quasiperiodic oscillation and restoring the dominance of the nonlinearities on the system’s response.

Coexistence of Hidden Attractors in the Smooth Cubic Chua’s Circuit with Two Stable Equilibria

Since the invention of Chua’s circuit, numerous generalizations based on the substitution of the nonlinear function have been reported. One of the generalizations is obtained by substituting cubic nonlinearity for piece-wise linear (PWL) nonlinearity. Although hidden chaotic attractors with a PWL nonlinearity have been discovered in the classical Chua’s circuit, chaotic attractors with a smooth cubic nonlinearity have long been known as self-excited attractors. Through a systematically exhaustive computer search, this paper identifies coexisting hidden attractors in the smooth cubic Chua’s circuit. Either self-excited or coexisting hidden attractors are now possible in the smooth cubic Chua’s circuit with algebraically elegant values of both initial points and system parameters. The newly found coexisting attractors exhibit an inversion symmetry. Both initial points and system parameters are equally required to localize hidden attractors. Basins of attraction of individual equilibria are illustrated and clearly show critical areas of multistability where a tiny drift in an initial point potentially induces jumps among different basins of attraction and coexisting states. Such multistability poses potential threats to engineering applications. The dynamical regions of hidden and self-excited attractors, and areas of stability of equilibria, are illustrated against two parameter spaces. Both illustrations reveal that two nonzero equilibrium points of the smooth cubic Chua’s circuit have a transition from unstable to stable equilibrium points, leading to generations of self-excited and hidden attractors simultaneously.

A small-scale agent-based model of institutional and technological change

This article develops a small-scale evolutionary agent-based model to investigate the interplay between heterogeneous agents, institutions and technological change. By acknowledging the concept of behavioural dispositions, we differentiate between changers, neutrals, and deniers. The composition of the population is endogenously determined taking into account that reasoning is context-dependent. As we increase the degree of interaction between agents, a bi-modal distribution with two different basins of attraction emerges: one around an equilibrium with the majority of the population supporting innovative change, and another with most agents being suspicious of innovation. Neutral agents play an important role as an element of resilience. Conditional on their share in equilibrium, an increase in the response of the respective probability functions to growth results in a super-critical Hopf-bifurcation, followed by the emergence of persistent fluctuations. Numerical experiments on the basin of attraction also reveal the birth of an additional periodic attractor. A long-run cycle of technological and institutional change may coexist with locally stable fixed points. Our results indicate that economies are more likely to be path-dependent than what conventional approaches usually admit.

A Fractional-Order Sinusoidal Discrete Map.

In this paper, a novel fractional-order discrete map with a sinusoidal function possessing typical nonlinear features, including chaos and bifurcations, is proposed. Firstly, the basic properties involving the stability of the equilibrium points and the symmetry of the map are studied by theoretical analysis. Secondly, the dynamics of the map in commensurate-order and incommensurate-order cases with initial conditions belonging to different basins of attraction is investigated by numerical simulations. The bifurcation types and influential parameters of the map are analyzed via nonlinear tools. Hopf, period-doubling, and symmetry-breaking bifurcations are observed when a parameter or an order is varied. Bifurcation diagrams and maximum Lyapunov exponent spectrums, with both a variation in a system parameter and an order or two orders, are shown in a three-dimensional space. A comparison of the bifurcations in fractional-order and integral-order cases shows that the variation in an order has no effect on the symmetry-breaking bifurcation point. Finally, the heterogeneous hybrid synchronization of the map is realized by designing suitable controllers. It is worth noting that the increase in a derivative order can promote the synchronization speed for the fractional-order discrete map.

Iterated nth order nonlinear quantum dynamics with mixed initial states

Nonlinear quantum transformations play an important role in quantum state distillation, aiding quantum communication schemes. The simplest of these involve quadratic rational maps for qubits, higher order versions have not been explored in detail. We consider such iterated nonlinear protocols for qubits, involving a generalized CNOT, a measurement, and a single-qubit Hadamard gate processing n inputs in identical quantum states. To characterize the global properties of these protocols, we determine the fixed points and relevant fixed cycles. We show that a repelling mixed fixed point exists for all orders, marking a phase transition related to the disappearance of the fractalness of borders of different basins of attraction.

Sub-Synchronous Oscillations and Transient Stability of Islanded Microgrid

Sustained sub-synchronous oscillations are observed in an islanded microgrid consisting of grid-forming converters under a large transient disturbance. Such oscillations are studied in terms of various types of hidden attractors, namely, hidden periodic orbit, hidden quasi-periodic orbit, and hidden chaotic attractor, which are shown to coexist with the stable equilibrium point, share different basins of attraction, and be transformed mutually for different parameter values. Moreover, the basins of attraction have fractal boundaries, making the nearby trajectories unpredictable. The transient stability of the islanded microgrid can be improved by eliminating the hidden attractors through adjusting control parameters and triggering a homoclinic bifurcation. Furthermore, transient disturbances can be mitigated by protecting critical converters and accelerating the restarting process. Besides, the emergence of sub-synchronous oscillations can be eliminated through tripping off one or more specific converters. Full-circuit cycle-by-cycle simulations are performed for islanded microgrids comprising i) 5 grid-forming converters, ii) 5 grid-forming converters and a synchronous generator, and iii) 30 grid-forming converters to verify the analytical findings.

Wada index based on the weighted and truncated Shannon entropy

The Wada index based on the weighted and truncated Shannon entropy is presented in this paper. The proposed Wada index can detect if a given basin boundary is a Wada boundary. Moreover, the Wada index does represent the number and the distribution of different colors (attractors) in the two-dimensional phase space of initial conditions. The Wada index is based on the standard box counting algorithm, but computations in each box are based on the weighted and truncated Shannon entropy. That makes the algorithm for the computation of the Wada index conveniently applicable for different basins of attraction represented as color digital images. Computational experiments are used to demonstrate the advantages of the proposed index and compare it with other existing techniques and algorithms for the numerical analysis of Wada boundaries.

How Do You Feel About Going <i>Green</i>? Modelling Environmental Sentiments in a Growing Open Economy

Contrary to the near global consensus among the scientific community, public perceptions of climate change are known to differ between nations and to have fluctuated over time. To study such a stylised fact, this article develops a small-scale agent-based model that allows for feedback effects between sentiments, environmental regulation and macroeconomic outcomes in an open economy set-up. Conditional on the level of interaction between agents, two different basins of attraction emerge: one with the majority of the population supporting climate change mitigation policies, and another with most agents opposing environmental regulation. We demonstrate that complex dynamics might occur via a sequence of period-doubling bifurcations. Employment series are more persistent than sentiments, resulting in relatively high volatility in the latter and smooth long-waves in the labour market. A sufficiently robust response of sentiments to emissions may lead to the disappearance of the Pareto inferior equilibria, allowing for a unique green steady-state.

Variability in the noise-induced modes of climate dynamics

New features of noise-induced climate variability are revealed on the basis of the three-dimensional model derived by Saltzman and Maasch. It is shown that the climate system can be highly noise excitable and it possesses the large-amplitude fluctuations even in those regions where its akin deterministic model does not contain any self-sustained oscillations. Intermittency in small- and large amplitude climate fluctuations between different basins of attraction of a limit cycle and stable equilibria substantially influencing the climate state (from warm to cold and vice versa) are found at various noise intensities. Suddenly occurring jumps between the basins of attraction of two stable equilibria corresponding to the warm and cold climate states are statistically confirmed under a certain diapason of noise intensities. The climate system undergoes transitions between its equilibria in the presence of noise in its prognostic variables. In addition, such transitions become more likely with increasing the noise intensity.

Emergence of a square chaotic attractor through the collision of heteroclinic orbits

In this work we introduce a square chaotic attractor based on the collision of two heteroclinic orbits. Before the collision, the system presents the coexistence of two double scroll attractors that are generated via piecewise linear (PWL) systems that deal with two saddle-foci equilibria of different classes, i.e. the dimensions of the unstable and stable manifolds of one of the equilibrium points are one and two, respectively, and vice versa for the unstable and stable manifolds associated with the second equilibrium point. The new class is constructed based on the existence of a heteroclinic loop by linear affine systems with two saddle-focus equilibrium points of different types. The chaotic behavior of the proposed system is tested by the maximum Lyapunov exponent and the 0–1 chaos test. This new class of PWL dynamical systems exhibits different behaviors and allows the generation of different basins of attraction for the coexistence of two or more attractors. Therefore, a mechanism to establish bistability is provided and how it can be controlled (i.e. bistability annihilation) via varying system’s parameters is demonstrated.

Reversals of period doubling, coexisting multiple attractors, and offset boosting in a novel memristive diode bridge-based hyperjerk circuit

In this paper, a new memristive diode bridge-based RC hyperjerk circuit is proposed. This new memristive hyperjerk oscillator (MHO) is obtained from the autonomous 4-D hyperjerk circuit (Leutcho et al. in Chaos Solitons Fractals 107:67–87, 2018) by replacing the nonlinear component (formed by two antiparallel diodes) with a first order memristive diode bridge. The circuit is described by a fifth-order continuous time autonomous (‘elegant’) hyperjerk system with smooth nonlinearities. The dynamics of the system is investigated in terms of equilibrium points and stability, phase portraits, bifurcation diagrams and two-parameter Lyapunov exponents diagrams. The numerical analysis of the model reveals interesting behaviors such as period-doubling, chaos, offset boosting, symmetry recovering crisis, antimonotonicity (i.e. concurrent creation and destruction of periodic orbits) and several coexisting bifurcations as well. One of the most attractive features of the new MHO considered in this work is the presence of several coexisting attractors (e.g. coexistence of two, three, four, five, six, seven, or nine attractors) for some suitable sets of system parameters, depending on the choice of initial conditions. Accordingly, the distribution of initial conditions related to each coexisting attractor is computed to highlight different basins of attraction. Laboratory experimental measurements are carried out to verify the theoretical analysis.

Distributional analysis for understanding geochemical processes affecting ground and surficial waters in different geological conditions

The aim of this contribution is to investigate whether the study of frequency distribution, representing chemical reactions or geochemical processes, is able to provide useful information on the dynamics affecting ground and surficial waters. To reach this goal we started to explore the behaviour of the main solutes and common contaminants, by means of distributional analysis. In fact, the shape of the frequency distribution in geochemical variables reveals important information about the governing dynamics of the system that gave birth to the distribution. Moreover, the distribution shape, when plurimodality is pointed out, can be used to approximate the shape of different basins of attraction of the concentration valuesand this allows us to make helpful observations about the resilience of the system taken into account. Two application examples are presented, the first one for Perchloroethylene (PCE) concentration in groundwater of Firenze-Prato-Pistoia (Tuscany Region) alluvial plain and the second one for the chemistry of the surficial waters of the Alpine region.

Mechanism of solitary state appearance in an ensemble of nonlocally coupled Lozi maps

We study the peculiarities of the solitary state appearance in the ensemble of nonlocally coupled chaotic maps. We show that nonlocal coupling and features of the partial elements lead to arising of multistability in the system. The existence of solitary state is caused by formation of two attractive sets with different basins of attraction. Their basins are analysed depending on coupling parameters.

Effects of different initial conditions on the emergence of chimera states

Chimeras are fascinating spatiotemporal states that emerge in coupled oscillators. These states are characterized by the coexistence of coherent and incoherent dynamics, and since their discovery, they have been observed in a rich variety of different systems. Here, we consider a system of non-locally coupled three-dimensional dynamical systems, which are characterized by the coexistence of fixed-points, limit cycles, and strange attractors. This coexistence creates an opportunity to study the effects of different initial conditions – from different basins of attraction – on the emergence of chimera states. By choosing initial conditions from different basins of attraction, and by varying also the coupling strength, we observe different spatiotemporal solutions, ranging from chimera states to synchronous, imperfect synchronous, and asynchronous states. We also determine conditions, in dependence on the basins of attraction, that must be met for the emergence of chimera states.

PageRank centrality for performance prediction: the impact of the local optima network model

A local optima network (LON) compresses relevant features of fitness landscapes in a complex network, where nodes are local optima and edges represent transition probabilities between different basins of attraction. Previous work has found that the PageRank centrality of local optima can be used to predict the success rate and average fitness achieved by local search based metaheuristics. Results are available for LONs where edges describe either basin transition probabilities or escape edges. This paper studies the interplay between the type of LON edges and the ability of the PageRank centrality for the resulting LON to predict the performance of local search based metaheuristics. It finds that LONs are stochastic models of the search heuristic. Thus, to achieve an accurate prediction, the definition of the LON edges must properly reflect the type of diversification steps used in the metaheuristic. LONs with edges representing basin transition probabilities capture well the diversification mechanism of simulated annealing which sometimes also accepts worse solutions that allow the search process to pass between basins. In contrast, LONs with escape edges capture well the diversification step of iterated local search, which escapes from local optima by applying a larger perturbation step.

Bifurcation Analysis of the Yamada Model for a Pulsing Semiconductor Laser with Saturable Absorber and Delayed Optical Feedback

Semiconductor lasers exhibit a wealth of dynamics, from emission of a constant beam of light to periodic oscillations and excitability. Self-pulsing regimes, where the laser periodically releases a short pulse of light, are particularly interesting for many applications, from material science to telecommunications. Self-pulsing regimes need to produce pulses very regularly and, as such, they are also known to be particularly sensitive to perturbations, such as noise or light injection. We investigate the effect of delayed optical feedback on the dynamics of a self-pulsing semiconductor laser with saturable absorber (SLSA). More precisely, we consider the Yamada model with delay---a system of three delay-differential equations (DDEs) for two slow and one fast variables---which has been shown to reproduce accurately self-pulsing features as observed in SLSA experimentally. This model is also of broader interest because it is quite closely related to mathematical models of other self-pulsing systems, such as excitable spiking neurons. We perform a numerical bifurcation analysis of the Yamada model with delay, where we consider the feedback delay, the feedback strength, and the strength of pumping as bifurcation parameters. We find a rapidly increasing complexity of the system dynamics when the feedback delay is increased from zero. In particular, there are new feedback-induced dynamics: stable quasi-periodic oscillations on tori, as well as a large degree of multistability, with up to five pulse-like stable periodic solutions with different amplitudes and repetition rates. An attractor map in the plane of perturbations on the gain and intensity reveals a Cantor-set--like, intermingled structure of the different basins of attraction. This suggests that, in practice, the multistable laser is extremely sensitive to small perturbations.

Digit replacement: A generic map for nonlinear dynamical systems.

A simple discontinuous map is proposed as a generic model for nonlinear dynamical systems. The orbit of the map admits exact solutions for wide regions in parameter space and the method employed (digit manipulation) allows the mathematical design of useful signals, such as regular or aperiodic oscillations with specific waveforms, the construction of complex attractors with nontrivial properties as well as the coexistence of different basins of attraction in phase space with different qualitative properties. A detailed analysis of the dynamical behavior of the map suggests how the latter can be used in the modeling of complex nonlinear dynamics including, e.g., aperiodic nonchaotic attractors and the hierarchical deposition of grains of different sizes on a surface.

Memory recall and spike-frequency adaptation.

The brain can reproduce memories from partial data; this ability is critical for memory recall. The process of memory recall has been studied using autoassociative networks such as the Hopfield model. This kind of model reliably converges to stored patterns that contain the memory. However, it is unclear how the behavior is controlled by the brain so that after convergence to one configuration, it can proceed with recognition of another one. In the Hopfield model, this happens only through unrealistic changes of an effective global temperature that destabilizes all stored configurations. Here we show that spike-frequency adaptation (SFA), a common mechanism affecting neuron activation in the brain, can provide state-dependent control of pattern retrieval. We demonstrate this in a Hopfield network modified to include SFA, and also in a model network of biophysical neurons. In both cases, SFA allows for selective stabilization of attractors with different basins of attraction, and also for temporal dynamics of attractor switching that is not possible in standard autoassociative schemes. The dynamics of our models give a plausible account of different sorts of memory retrieval.

Quantifying the landscape and kinetic paths for epithelial-mesenchymal transition from a core circuit.

Epithelial-mesenchymal transition (EMT), as a crucial process in embryonic development and cancer metastasis, has been investigated extensively. However, how to quantify the global stability and transition dynamics for EMT under fluctuations remains to be elucidated. Starting from a core EMT genetic circuit composed of three key proteins or microRNAs (microRNA-200, ZEB and SNAIL), we uncovered the potential landscape for the EMT process. Three attractors emerge from the landscape, which correspond to epithelial, mesenchymal and partial EMT states respectively. Based on the landscape, we analyzed two important quantities of the EMT system: the barrier heights between different basins of attraction that describe the degree of difficulty for EMT or backward transition, and the mean first passage time (MFPT) that characterizes the kinetic transition rate. These quantities can be harnessed as measurements for the stability of cell types and the degree of difficulty of transitions between different cell types. We also calculated the minimum action paths (MAPs) by path integral approaches. The MAP delineates the transition processes between different cell types quantitatively. We propose two different EMT processes: a direct EMT from E to P, and a step-wise EMT going through an intermediate state, depending on different extracellular environments. The landscape and kinetic paths we acquired offer a new physical and quantitative way for understanding the mechanisms of EMT processes, and indicate the possible roles for the intermediate states.

Testing for Basins of Wada.

Nonlinear systems often give rise to fractal boundaries in phase space, hindering predictability. When a single boundary separates three or more different basins of attraction, we say that the set of basins has theWada property and initial conditions near that boundary are even more unpredictable. Many physical systems of interest with this topological property appear in the literature. However, so far the only approach to study Wada basins has been restricted to two-dimensional phase spaces. Here we report a simple algorithm whose purpose is to look for the Wada property in a given dynamical system. Another benefit of this procedure is the possibility to classify and study intermediate situations known as partially Wada boundaries.