Multistable Systems Research Articles

Sampling can be faster than optimization

Optimization algorithms and Monte Carlo sampling algorithms have provided the computational foundations for the rapid growth in applications of statistical machine learning in recent years. There is, however, limited theoretical understanding of the relationships between these 2 kinds of methodology, and limited understanding of relative strengths and weaknesses. Moreover, existing results have been obtained primarily in the setting of convex functions (for optimization) and log-concave functions (for sampling). In this setting, where local properties determine global properties, optimization algorithms are unsurprisingly more efficient computationally than sampling algorithms. We instead examine a class of nonconvex objective functions that arise in mixture modeling and multistable systems. In this nonconvex setting, we find that the computational complexity of sampling algorithms scales linearly with the model dimension while that of optimization algorithms scales exponentially.

Humans Perceive Binocular Rivalry and Fusion in a Tristable Dynamic State

Human vision combines inputs from the two eyes into one percept. Small differences "fuse" together, whereas larger differences are seen "rivalrously" from one eye at a time. These outcomes are typically treated as mutually exclusive processes, with paradigms targeting one or the other and fusion being unreported in most rivalry studies. Is fusion truly a default, stable state that only breaks into rivalry for non-fusible stimuli? Or are monocular and fused percepts three sub-states of one dynamical system? To determine whether fusion and rivalry are separate processes, we measured human perception of Gabor patches with a range of interocular orientation disparities. Observers (10 female, 5 male) reported rivalrous, fused, and uncertain percepts over time. We found a dynamic "tristable" zone spanning from ∼25-35° of orientation disparity where fused, left-eye-, or right-eye-dominant percepts could all occur. The temporal characteristics of fusion and non-fusion periods during tristability matched other bistable processes. We tested statistical models with fusion as a higher-level bistable process alternating with rivalry against our findings. None of these fit our data, but a simple bistable model extended to have three states reproduced many of our observations. We conclude that rivalry and fusion are multistable substates capable of direct competition, rather than separate bistable processes.SIGNIFICANCE STATEMENT When inputs to the two eyes differ, they can either fuse together or engage in binocular rivalry, where each eye's view is seen exclusively in turn. Visual stimuli have often been tailored to produce either fusion or rivalry, implicitly treating them as separate mutually-exclusive perceptual processes. We have found that some similar-but-different stimuli can result in both outcomes over time. Comparing various simple models with our results suggests that rivalry and fusion are not independent processes, but compete within a single multistable system. This conceptual shift is a step toward unifying fusion and rivalry, and understanding how they both contribute to the visual system's production of a unified interpretation of the conflicting images cast on the retina by real-world scenes.

Perturbation Theory and Singular Perturbations for Input-to-State Multistable Systems on Manifolds

We consider the notion of input-to-state multistability, which generalizes input-to-state stability to nonlinear systems evolving on Riemannian manifolds and possessing a finite number of compact, globally attractive, invariant sets, and in addition satisfies a specific condition of acyclicity. We prove that a parameterized family of dynamical systems whose solutions converge to those of a limiting system inherits such input-to-state multistability property from the limiting system in a semiglobal practical fashion. A similar result is also established for singular perturbation models whose boundary-layer subsystem is uniformly asymptotically stable and whose reduced subsystem is input-to-state multistable. Known results in the theory of perturbations, singular perturbations, averaging, and highly oscillatory control systems are here generalized to the multistable setting by replacing the classical asymptotic stability requirement of a single invariant set with attractivity and acyclicity of a decomposable invariant one.

Predicting noise-induced critical transitions in bistable systems.

Critical transitions from one dynamical state to another contrasting state are observed in many complex systems. To understand the effects of stochastic events on critical transitions and to predict their occurrence as a control parameter varies are of utmost importance in various applications. In this paper, we carry out a prediction of noise-induced critical transitions using a bistable model as a prototype class of real systems. We find that the largest Lyapunov exponent and the Shannon entropy can act as general early warning indicators to predict noise-induced critical transitions, even for an earlier transition due to strong fluctuations. Furthermore, the concept of the parameter dependent basin of the unsafe regime is introduced via incorporating a suitable probabilistic notion. We find that this is an efficient tool to approximately quantify the range of the control parameter where noise-induced critical transitions may occur. Our method may serve as a paradigm to understand and predict noise-induced critical transitions in multistable systems or complex networks and even may be extended to a broad range of disciplines to address the issues of resilience.

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.

Multistable Cosine-Curved Dome System for Elastic Energy Dissipation

This paper presents a new energy dissipation system composed of multistable cosine-curved domes (CCD) connected in series. The system exhibits multiple consecutive snap-through and snap-back buckling behavior with a hysteretic response. The response of the CCDs is within the elastic regime and hence the system's original configuration is fully recoverable. Numerical studies and experimental tests were conducted on the geometric properties of the individual CCD units and their number in the system to examine the force–displacement and energy dissipation characteristics. Finite element analysis (FEA) was performed to simulate the response of the system to develop a multilinear analytical model for the hysteretic response that considers the nonlinear behavior of the system. The model was used to study the energy dissipation characteristics of the system. Experimental tests on 3D printed specimens were conducted to analyze the system and validate numerical results. Results show that the energy dissipation mainly depends on the number and the apex height-to-thickness ratio of the CCD units. The developed multilinear analytical model yields conservative yet accurate values for the dissipated energy of the system. The proposed system offered reliable high energy dissipation with a maximum loss factor value of 0.14 for a monostable (self-recoverable) system and higher for a bistable system.

Extreme multistable synchronisation in coupled dynamical systems

A rule for designing extreme multistable synchronised systems by coupling two identical dynamical systems has been proposed in this paper. The basic idea behind the proposed scheme is the existence of chaos in the coupled system in the presence of initial condition-dependent constants of motion. A new conjecture has been introduced according to which an extreme multistable synchronised system can be designed if all states of one system will synchronise with the corresponding states of the other system (of the two coupled systems) and the basin of the synchronised state depends on the difference between the initial conditions of the corresponding states of the individual systems. The proposed scheme has been illustrated with the help of coupled Rossler systems, coupled Henon maps and coupled logistic maps. Moreover, the existence of flip bifurcation with the variation of initial conditions has been shown analytically as well as numerically in the case of coupled Henon maps. Numerical results are reported to show the proficiency of the proposed scheme to design extreme multistable synchronisation behaviour. This work establishes a theoretical foundation for constructing extreme multistable synchronised continuous as well as discrete dynamical systems.

The Jacobi Stability of a Lorenz-Type Multistable Hyperchaotic System with a Curve of Equilibria

In this paper, a 4D Lorenz-type multistable hyperchaotic system with a curve of equilibria is investigated by using differential geometry method, i.e. with KCC-theory. Due to the deviation curvature tensor and its eigenvalues, the curve of equilibria of this hyperchaotic system is proved analytically to be Jacobi unstable under a certain parameter condition, and a periodic orbit of this system is proved numerically to be also Jacobi unstable. Furthermore, the dynamics of contravariant vector field near the curve of equilibria and the periodic orbit are studied, respectively, and their results comply absolutely with the above analysis of Jacobi stability.

Dynamics and Entropy Analysis for a New 4-D Hyperchaotic System with Coexisting Hidden Attractors.

This paper presents a new no-equilibrium 4-D hyperchaotic multistable system with coexisting hidden attractors. One prominent feature is that by varying the system parameter or initial value, the system can generate several nonlinear complex attractors: periodic, quasiperiodic, multiple topology chaotic, and hyperchaotic. The dynamics and complexity of the proposed system were investigated through Lyapunov exponents (LEs), a bifurcation diagram, a Poincaré map, and spectral entropy (SE). The simulation and calculation results show that the proposed multistable system has very rich and complex hidden dynamic characteristics. Additionally, the circuit of the chaotic system is designed to verify the physical realizability of the system. This study provides new insights into uncovering the dynamic characteristics of the coexisting hidden attractors system and provides a new choice for nonlinear control or chaotic secure communication technology.

Tailoring multistable vibrational energy harvesters for enhanced performance: theory and numerical investigation

This study exposes the analytical and numerical analyses of multistable systems for energy harvesting purposes. More specifically, this paper aims at providing appropriate conditions for ensuring equal potential barrier to go from one well to another whatever the order of multistability. This therefore allows optimal operations through either potential barrier lowering or vibration magnitude increase. Then, such analytical and numerical results are incorporated into a general dynamic model and evaluated. Results show a significant magnification of the frequency bandwidth while keeping the same maximal velocity magnitude. Hence, such a unified approach would permit designing efficient vibrational energy harvester working on a wide frequency band at low excitation levels.

A novel parameter-induced adaptive stochastic resonance system based on composite multi-stable potential model

Stochastic resonance (SR) is used widely as a weak signal detection method by using noise in many fields. In order to improve the weak signal processing capability of SR, a novel composite multi-stable model is proposed, which is constructed by the joint of the tristable model and the Gaussian Potential (GP) model. The SR system based on this model is constructed and the signal-to-noise ratio (SNR) is regarded as the index to measure the SR effect. The differential brain storm optimization (DBSO) algorithm is used to optimize the system parameters collaboratively to achieve parameter-induced adaptive SR. The influences of the system parameters V and R and the noise intensity D on the output response of SR system are analyzed under Gaussian white noise and α stable noise environments, and the advantages of the composite multi-stable SR system over the traditional tristable system are verified. For different levels of weak signals, the output performances of SR systems based on composite multi-stable model, traditional tristable model, composite tristable model are compared and analyzed. The results prove that the proposed model has better performance. Meanwhile, the adaptive detection of the multiple high-frequency weak signal is realized using the composite multi-stable SR system. The simulation results show that the proposed system has strong weak signal processing capability and good immunity to noise types, which widens the application range of SR in practical engineering.

Dynamical analysis of a new multistable chaotic system with hidden attractor: Antimonotonicity, coexisting multiple attractors, and offset boosting

Analyzing chaotic systems with coexisting and hidden attractors has been receiving much attention recently. In this article, we analyze a four dimensional chaotic system which has a plane as the equilibrium points. Also this system is of the group of systems that have coexisting attractors. First, the system is introduced and then stability analysis, bifurcation diagram and Largest Lyapunov exponent of this system are presented as methods to analyze the multistability of the system. These methods reveal that in some ranges of the parameter, this chaotic system has three different types of coexisting attractors, chaotic, stable node and limit cycle. Some interesting dynamics properties such as reversals of period doubling bifurcation and offset boosting are also presented.

Approximating the energy landscape of a two-dimensional bistable gene autoregulation model by separating slow and fast dynamics.

The energy landscape is widely used to quantify the stability of multistable nonlinear systems, such as bistable gene regulation networks. In physics, the potential can be obtained through integration only for gradient systems. However, multidimensional nonlinear systems are often nongradient, for which the potential is calculated by decomposing the dynamics to gradient and nongradient parts. This potential is then called a quasipotential. Given that one-dimensional (1D) systems can be regarded as gradient systems, we attempt to separate the two-dimensional (2D) system into two 1D systems working on distinct timescales, and the potential can be easily calculated for the two 1D systems separately. This method is used in this study to estimate the energy landscape of a two-variable gene autoregulation model. This elegant and comprehensive method is accessible for 2D nonlinear systems in which the dynamics can be divided into slow and fast parts.

Hybrid State Variable Incremental Integral for Reconstructing Extreme Multistability in Memristive Jerk System with Cubic Nonlinearity

Memristive system with infinitely many equilibrium points has attracted much attention for the generation of extreme multistability, whose initial‐dependent dynamics can be interpreted in a reduced‐order model through incremental integral transformation of state variables. But, the memristive system with any extra nonlinear terms besides the memristor ones cannot be handled directly using this method. In addition, the transformed state variables could be divergent due to the asymmetry of the original system. To solve these problems, a hybrid state variable incremental integral (HSVII) method is proposed in this paper. With this method, the extreme multistability in a four‐dimensional (4D) memristive jerk system with cubic nonlinearity is successfully reconstituted in a three‐dimensional (3D) model and the divergent state variables are eliminated through ingenious linear state variable mapping. Thus, mechanism analysis and physical control of the special extreme multistability can readily be performed. A hardware circuit is finally designed and fabricated, and the theoretical and numerical results are verified by the experimental measurements. It is demonstrated that this HSVII method is effective for the analysis of multistable system with high‐order nonlinearities.

Generating extremely multistable systems based on Lurie form systems

Generating extremely multistable systems based on Lurie form systems

Influence of transient escape and added load noise on the dynamic integrity of multistable systems

Concepts and tools of global dynamics play a meaningful role in enhancing the engineering design and safety of multistable structural systems in a dynamic environment. Basins of attraction topology can vary meaningfully as a function of system parameters, and the basin boundaries can be smooth or fractal. Fractal boundaries have important practical consequences as they lead to uncertainty with respect to the outcome of a given initial condition, thus reducing the system safety, which can be quantified through different dynamic integrity measures and their variation with a control parameter. In the present paper two key issues influencing the integrity measures and basins’ erosion profiles are addressed: transient escape and added random noise. Transient escape is a significant issue in many structural systems since it may lead to unwanted stresses and strains or exceed some safe design requirements in terms of maximum displacements and velocities. In turn, the random noise may increase the uncertainty of the systems, thus decreasing their practical safety and escape load. As a case study, the bistable von Mises truss is considered and the effects of transient escape and added random noise on the global dynamics of the harmonically excited truss are investigated in detail, clarifying their influence on the system safety as evaluated with different integrity measures.

Reprogramming cooperative monotone dynamical systems.

Multistable dynamical systems are ubiquitous in nature, especially in the context of regulatory networks controlling cell fate decisions, wherein stable steady states correspond to different cell phenotypes. In the past decade, it has become experimentally possible to "reprogram" the fate of a cell by suitable externally imposed input stimulations. In several of these reprogramming instances, the underlying regulatory network has a known structure and often it falls in the class of cooperative monotone dynamical systems. In this paper, we therefore leverage this structure to provide concrete guidance on the choice of inputs that reprogram a cooperative dynamical system to a desired target steady state. Our results are parameter-independent and therefore can serve as a practical guidance to cell-fate reprogramming experiments.

Detecting early-warning signals in periodically forced systems with noise.

Early-warning signals for imminent regime shifts in multi-stable systems are highly desirable because it is often difficult to revert a system to the previous state once a transition has occurred. In this paper, two indicators, the phase lag and amplitude difference of the system's response, are extended to detect early-warning signals of a periodically driven, bistable complex system with noise. Our results show that both indicators can announce a regime shift of a complex system with small noise, namely, the critical point of the regime shift near a bifurcation point of the corresponding deterministic system. However, they fail to early indicate the regime shift in the case of large noise where the shift is far from the original bifurcation point. Based on the moment-expanding scheme, we reduce a large noise to a small one, and then both indicators work well again. We illustrate this approach via a parameterized lake eutrophication model verified by data. The regime shift to eutrophication could be detected in advance by studying the phase lag and amplitude difference of phosphorus concentrations. A basic statistical test is performed for the robustness of the proposed indicators. This approach provides a theoretical basis to prevent ecological environment deteriorations.

New topological tool for multistable dynamical systems.

We introduce a new method for investigation of dynamical systems which allows us to extract as much information as possible about potential system dynamics, based only on the form of equations describing it. The discussed tool of critical surfaces, defined by the zero velocity (and/or) acceleration field for particular variables of the system is related to the geometry of the attractors. Particularly, the developed method provides a new and simple procedure allowing to localize hidden oscillations. Our approach is based on the dimension reduction of the searched area in the phase space and has an advantage (in terms of complexity) over standard procedures for investigating full-dimensional space. The two approaches have been compared using typical examples of oscillators with hidden states. Our topological tool allows us not only to develop alternate ways of extracting information from the equations of motion of the dynamical system, but also provides a better understanding of attractors geometry and their capturing in complex cases, especially including multistable and hidden attractors. We believe that the introduced method can be widely used in the studies of dynamical systems and their applications in science and engineering.

Impulsive control of a nonlinear dynamical network and its application to biological networks.

The control of nonlinear dynamical systems is always a notable problem in science. According to control theory, suitable inputs for a controllable dynamical system are critical. Previous studies have shown some principles to determine control nodes and design control function. In this work, we propose a new control strategy of nonlinear systems by constructing impulsive control functions, i.e., we can realize the transition from an undesired state to a desired one by controlling appropriately chosen nodes in a discrete manner. In order to demonstrate the effectiveness of the strategy, we apply it to two biological networks: the epithelial-mesenchymal transition (EMT) network and the Notch1-Dll1-Jag1 signaling pathway. The strategy can not only be used to guide pharmacological design in a more feasible form but can also be applied into the fields of biological, medical and other multistable dynamical systems.