Multistability Phenomena Research Articles

Analysis on Multistable Motion Characteristics of Supercavitating Vehicle

Due to complex underwater environment, when the initial condition of launching is subjected to low external disturbance, the motion trace of a supercavitating vehicle might display many different motion states during underwater navigation. With the aim of addressing this problem, based on the dynamic map, in the present work the multistable phenomena of attractor coexistence of the supercavitating vehicle system under various initial conditions were analyzed and the initial condition effects on the multistable motion characteristics were investigated through the domains of attraction, time, and frequency. The results demonstrated that, unlike the ordinary dynamic systems, a supercavitating vehicle demonstrates multistable phenomena, such as the coexistence of the stable equilibrium point and the limit cycle and the coexistence of the limit cycle and the chaotic attractor, along with the coexistence of diversified limit cycles; under fixed system parameters, as the initial condition of launching varied, the vehicle displayed various motion states; in engineering practices, the initial condition of launching could be adjusted according to the domain of attraction, in order for the vehicle motion stability to be enhanced.

Information and Self-Organization

The process of “self-organization” takes place in open and complex systems that acquire spatio-temporal or functional structures without specific ordering instructions from the outside. [...]

Dynamics of a multiplier–accelerator model with nonlinear investment function

In this paper, we show how a rich variety of dynamics may arise in a simple multiplier–accelerator model when a nonlinearity is introduced in the investment function. A specific sigmoidal functional form is used to model investments with respect to the variation in national income, in order to bound the level of investments. In fact, due to obvious material constraints, business strategies cannot sustain infinite investments. With the aid of analytical and numerical tools, we investigate the stability conditions, bifurcations, as well as periodic and chaotic dynamics for different specifications of the model that may include or not an endogenous government expenditure. We obtain some comparative statics results that shed light on the stabilizing or destabilizing role of the various parameters in the model. Globally, we study multistability phenomena, i.e., the coexistence of different kinds of attractors with a rich and complex dynamic structure.

Hidden extreme multistability in memristive hyperchaotic system

By utilizing a memristor to substitute a coupling resistor in the realization circuit of a three-dimensional chaotic system having one saddle and two stable node-foci, a novel memristive hyperchaotic system with coexisting infinitely many hidden attractors is presented. The memristive system does not display any equilibrium, but can exhibit hyperchaotic, chaotic, and periodic dynamics as well as transient hyperchaos. In particular, the phenomenon of extreme multistability with hidden oscillation is revealed and the coexistence of infinitely many hidden attractors is observed. The results illustrate that the long term dynamical behavior closely depends on the memristor initial condition thus leading to the emergence of hidden extreme multistability in the memristive hyperchaotic system. Additionally, hardware experiments and PSIM circuit simulations are performed to verify numerical simulations.

Optical bistability and multistability in four-level atoms with a diamond configuration

The phenomena of optical bistability and multistability are investigated for four-level atoms with a diamond configuration trapped in a unidirectional optical ring cavity, driven by a probe as well as three coupling fields. Moreover, the optical properties of the atoms are studied considering the spontaneously generated coherence and Rabi frequencies. The results indicate that the optical bistability and multistability of the corresponding atom-cavity system can be controlled by two spontaneously generated coherences in upper and lower decay channels and the Rabi frequency of the coupling fields.

Complex dynamics and multistability with increasing rationality in market games

In this work we study oligopoly models in which firms adopt decision mechanisms based on best response techniques with different rationality degrees. Firms are also assumed to face resource or financial constraints in adjusting their production levels, so that, from time to time, they can only increase or decrease their strategy by a bounded quantity. We consider different families of oligopolies of generic sizes, characterized by heterogeneous compositions with respect to the rationality degrees of firms. We analytically study the local stability of the equilibrium depending on the oligopoly size and composition and through numerical simulations we investigate the possible dynamics arising when trajectories do not converge toward the equilibrium. We show that in this case complex dynamics can arise, and this is due to both the loss of stability of the equilibrium and to the emergence of multiple attractors, with the stable steady state coexisting with a different, periodic or chaotic, attractor. In particular, we show that multistability phenomena occur when the overall degree of rationality of the oligopoly is increased. Finally, we investigate the effect of non-convergent dynamics on the realized profits.

Quasiperiodicity and Chaos in a Generalized Nosé–Hoover Oscillator

This paper reports on an investigation of the two-dimensional parameter-space of a generalized Nosé–Hoover oscillator. It is a mathematical model of a thermostated harmonic oscillator, which consists of a set of three autonomous first-order nonlinear ordinary differential equations. By using Lyapunov exponents to numerically characterize the dynamics of the model at each point of this parameter-space, it is shown that dissipative quasiperiodic structures are present, embedded in a chaotic region. The same parameter-space is also used to confirm the multistability phenomenon in the investigated mathematical model.

On the Dynamics of Chua’s oscillator with a smooth cubic nonlinearity: occurrence of multiple attractors

Chua’s circuit is one of the well-known nonlinear circuits which have been used to study a rich variety of nonlinear dynamic behaviors such as bifurcation, chaos, and routes to chaos. In this work, I consider the dynamics of Chua’s circuit with a smooth cubic nonlinearity. The dynamics of the model is investigated using standard nonlinear analysis techniques including time series, bifurcation diagrams, phase space trajectories plots, Lyapunov exponents, and basins of attraction. Both period-doubling and crisis routes to chaos are reported. One of the major results of this work is the numerical finding of a parameter region in which Chua’s circuit experiences multiple attractors’ behavior (i.e., coexistence of four different periodic and chaotic attractors). This phenomenon was not reported previously in the Chua’s circuit (despite the huge amount of related research works) and thus represents an enriching contribution to the understanding of the dynamics of Chua’s oscillator. Basins of attraction of various coexisting attractors are depicted showing complex basin boundaries. The results obtained in this work let us conjecture that there are still some unknown and striking behaviors of Chua’s oscillator (e.g., the phenomenon of extreme multistability, i.e., infinitely many attractors) that need to be uncovered.

Collective Activity of Many Bistable Assemblies Reproduces Characteristic Dynamics of Multistable Perception.

The timing of perceptual decisions depends on both deterministic and stochastic factors, as the gradual accumulation of sensory evidence (deterministic) is contaminated by sensory and/or internal noise (stochastic). When human observers view multistable visual displays, successive episodes of stochastic accumulation culminate in repeated reversals of visual appearance. Treating reversal timing as a "first-passage time" problem, we ask how the observed timing densities constrain the underlying stochastic accumulation. Importantly, mean reversal times (i.e., deterministic factors) differ enormously between displays/observers/stimulation levels, whereas the variance and skewness of reversal times (i.e., stochastic factors) keep characteristic proportions of the mean. What sort of stochastic process could reproduce this highly consistent "scaling property?" Here we show that the collective activity of a finite population of bistable units (i.e., a generalized Ehrenfest process) quantitatively reproduces all aspects of the scaling property of multistable phenomena, in contrast to other processes under consideration (Poisson, Wiener, or Ornstein-Uhlenbeck process). The postulated units express the spontaneous dynamics of attractor assemblies transitioning between distinct activity states. Plausible candidates are cortical columns, or clusters of columns, as they are preferentially connected and spontaneously explore a restricted repertoire of activity states. Our findings suggests that perceptual representations are granular, probabilistic, and operate far from equilibrium, thereby offering a suitable substrate for statistical inference. Spontaneous reversals of high-level perception, so-called multistable perception, conform to highly consistent and characteristic statistics, constraining plausible neural representations. We show that the observed perceptual dynamics would be reproduced quantitatively by a finite population of distinct neural assemblies, each with locally bistable activity, operating far from the collective equilibrium (generalized Ehrenfest process). Such a representation would be consistent with the intrinsic stochastic dynamics of neocortical activity, which is dominated by preferentially connected assemblies, such as cortical columns or clusters of columns. We predict that local neuron assemblies will express bistable dynamics, with spontaneous active-inactive transitions, whenever they contribute to high-level perception.

From Necker Cubes to Polyrhythms: Fostering a Phenomenological Attitude in Music Education

Phenomenology is explored as a way of helping students and educators open up to music as a creative and transformative experience. I begin by introducing a simple exercise in experimental phenomenology involving multi-stable visual phenomena that can be explored without the use of complex terminology. Here, I discuss how the “phenomenological attitude” may foster a deeper appreciation of the structure of consciousness, as well as the central role the body plays in how we experience and form understandings of the worlds we inhabit. I then explore how the phenomenological attitude may serve as a starting point for students and teachers as they begin to reflect on their involvement with music as co-investigators. Here I draw on my teaching practice as a percussion and drum kit instructor, with a special focus on multi-stable musical phenomena (e.g., African polyrhythm). To conclude, I briefly consider how the phenomenological approach might be developed beyond the practice room to examine music’s relationship to the experience of culture, imagination and “self.”

Opinion evolution and rare events in an open community

Abstract There are many multi-stable phenomena in society. To explain these multi-stable phenomena, we have studied opinion evolution in an open community. We focus on probability of transition (or the mean transition time) that the system transfer from one state to another. We suggest a bistable model to provide an interpretation of these phenomena. The quasi-potential method that we used is the most important method to calculate the transition time and it can be used to determine the whole probability density. We study the condition of bistability and then discuss rare events in a multi-stable system. In our model, we find that two parameters, “temperature” and “persuading intensity,” influence the behavior of the system; a suitable “persuading intensity” and low “temperature” make the system more stable. This means that the transition rarely happens. The asymmetric phenomenon caused by “public-opinion” is also discussed.

Multistability and dynamic transitions of intracellular Min protein patterns.

Cells owe their internal organization to self‐organized protein patterns, which originate and adapt to growth and external stimuli via a process that is as complex as it is little understood. Here, we study the emergence, stability, and state transitions of multistable Min protein oscillation patterns in live Escherichia coli bacteria during growth up to defined large dimensions. De novo formation of patterns from homogenous starting conditions is observed and studied both experimentally and in simulations. A new theoretical approach is developed for probing pattern stability under perturbations. Quantitative experiments and simulations show that, once established, Min oscillations tolerate a large degree of intracellular heterogeneity, allowing distinctly different patterns to persist in different cells with the same geometry. Min patterns maintain their axes for hours in experiments, despite imperfections, expansion, and changes in cell shape during continuous cell growth. Transitions between multistable Min patterns are found to be rare events induced by strong intracellular perturbations. The instances of multistability studied here are the combined outcome of boundary growth and strongly nonlinear kinetics, which are characteristic of the reaction–diffusion patterns that pervade biology at many scales.

Endogenous evolution of heterogeneous consumers preferences: Multistability and coexistence between groups

We propose an exchange economy evolutionary model with agents heterogeneous in the structure of preferences. Assuming that the share updating mechanism is non-monotone in the calorie intake, we find multistability phenomena involving equilibria characterized by the coexistence of heterogeneous agents.

Basin stability in delayed dynamics.

Basin stability (BS) is a universal concept for complex systems studies, which focuses on the volume of the basin of attraction instead of the traditional linearization-based approach. It has a lot of applications in real-world systems especially in dynamical systems with a phenomenon of multi-stability, which is even more ubiquitous in delayed dynamics such as the firing neurons, the climatological processes, and the power grids. Due to the infinite dimensional property of the space for the initial values, how to properly define the basin’s volume for delayed dynamics remains a fundamental problem. We propose here a technique which projects the infinite dimensional initial state space to a finite-dimensional Euclidean space by expanding the initial function along with different orthogonal or nonorthogonal basis. A generalized concept of basin’s volume in delayed dynamics and a highly practicable calculating algorithm with a cross-validation procedure are provided to numerically estimate the basin of attraction in delayed dynamics. We show potential applicabilities of this approach by applying it to study several representative systems of biological or/and physical significance, including the delayed Hopfield neuronal model with multistability and delayed complex networks with synchronization dynamics.

Bifurcation of Periodic Solutions of the Mackey– Glass Equation

We study the bifurcation of the equilibrium states of periodic solutions for the Mackey– Glass equation. This equation is considered as a mathematical model of changes in the density of white blood cells. The equation written in dimensionless variables contains a small parameter at the derivative, which makes it singular. We applied the method of uniform normalization, which allows us to reduce the study of the solutions behavior in the neighborhood of the equilibrium state to the analysis of the countable system of ordinary differential equations. We poot out the equations in ”fast” and ”slow” variables from this system. Equilibrium states of the ”slow” variables equations determine the periodic solutions. The analysis of equilibrium states allows us to study the bifurcation of periodic solutions depending on the parameters of the equation and their stability. The possibility of simultaneous bifurcation of a large number of stable periodic solutions is shown. This situation is called the multistability phenomenon.

Voltage Multistability and Pulse Emergency Control for Distribution System With Power Flow Reversal

High levels of penetration of distributed generation and aggressive reactive power compensation may result in the reversal of power flows in future distribution grids. The voltage stability of these operating conditions may be very different from the more traditional power consumption regime. This paper focuses on the demonstration of multistability phenomenon in radial distribution systems with reversed power flow where multiple stable equilibria coexist for the given set of parameters. The system may experience transitions between different equilibria after being subjected to disturbances such as short-term losses of distributed generation or transient faults. Convergence to an undesirable equilibrium places the system in an emergency or in extremis state. Traditional emergency control schemes are not capable of restoring the system if it gets entrapped in one of the low voltage equilibria. Moreover, undervoltage load shedding may have a reverse action on the system and can induce voltage collapse. We propose a novel pulse emergency control strategy that restores the system to the normal state without any interruption of power delivery. The results are validated with dynamic simulations of IEEE 13-bus feeder performed with SystemModeler software. The dynamic models can also be used for characterization of the solution branches via a novel approach, so-called the admittance homotopy power flow method.

Rhythmic Inhibition Allows Neural Networks to Search for Maximally Consistent States.

Gamma-band rhythmic inhibition is a ubiquitous phenomenon in neural circuits, yet its computational role remains elusive. We show that a model of gamma-band rhythmic inhibition allows networks of coupled cortical circuit motifs to search for network configurations that best reconcile external inputs with an internal consistency model encoded in the network connectivity. We show that Hebbian plasticity allows the networks to learn the consistency model by example. The search dynamics driven by rhythmic inhibition enable the described networks to solve difficult constraint satisfaction problems without making assumptions about the form of stochastic fluctuations in the network. We show that the search dynamics are well approximated by a stochastic sampling process. We use the described networks to reproduce perceptual multistability phenomena with switching times that are a good match to experimental data and show that they provide a general neural framework that can be used to model other perceptual inference phenomena.

Local Dynamics of the Two-Component Singular Perturbed Systems of Parabolic Type

This paper considers the behavior of solutions from the neighborhood of an equilibrium state of nonlinear two-component parabolic problems with diffusion matrixes of one or two eigenvalues to zero. It has been shown that problems related to stability have infinite dimension. Reported here is the development of an algorithm that constructs universal families of nonlinear boundary-value problems which do not contain small parameters and whose nonlocal dynamics describes local dynamics of original boundary-value problems. In addition, an exhaustive set of universal systems for two-component parabolic equations is presented. It is concluded that a hyper multistability phenomenon is one characteristic of these systems.

Introducing a price variation limiter mechanism into a behavioral financial market model.

In the present paper, we consider a nonlinear financial market model in which, in order to decrease the complexity of the dynamics and to achieve price stabilization, we introduce a price variation limiter mechanism, which in each period bounds the price variation so that the current price is forced to belong to a certain interval determined by the price realization in the previous period. More precisely, we introduce such mechanism into a financial market model in which the price dynamics are described by a sigmoidal price adjustment mechanism characterized by the presence of two asymptotes that bound the price variation and thus the dynamics. We show that the presence of our asymptotes prevents divergence and negativity issues. Moreover, we prove that the basins of attraction are complicated only under suitable conditions on the parameters and that chaos arises just when the price limiters are loose enough. On the other hand, for some suitable parameter configurations, we detect multistability phenomena characterized by the presence of up to three coexisting attractors.

A financial market model with endogenous fundamental values through imitative behavior.

In this paper, we propose a financial market model with heterogeneous speculators, i.e., optimistic and pessimistic fundamentalists that, respectively, overestimate and underestimate the true fundamental value due to ambiguity in the stock market, which prevents them from relying on the true fundamental value in their speculations. Indeed, we assume that agents use in its place fundamental values determined by an imitative process. Namely, in forming their beliefs, speculators consider the relative profits realized by optimists and pessimists and update their fundamental values proportionally to those relative profits. Moreover, differently from the majority of the literature on the topic, the stock price is determined by a nonlinear mechanism that prevents divergence issues. For our model, we study, via analytical and numerical tools, the stability of the unique steady state, its bifurcations, as well as the emergence of complex behaviors. We also investigate multistability phenomena, characterized by the presence of coexisting attractors.