Multistable Systems Research Articles

Bifurcation and multi-stability analysis of microwave engineering systems: Insights from the Burger–Fisher equation

The present article depicts the research being done on the microwave effect realized by the often-used Burger-Fisher equation. A particular view is on the emergence of multi-stability and the bifurcations associated with microwave engineering systems. A novel rational, trigonometric, and hyperbolic form of the traveling waves is furnished by the resolution of the non-linear issue with the aid of the advanced exp (−Ψ(η))-expansion function parameterization. Firstly, we will solve the exact form of the Burger-Fisher equation, which clarifies the behavior of the equation in different conditions. Subsequently, bifurcation analysis techniques will be employed to study the nuanced relationship between the system parameters and the emergence of multistability phenomena. Through our results, we exposed the complicated essential features of microwave physics and established crucial parameters that determine a system’s behavior. Next, we cover control principle issues and practical applications in microwave engineering problems. This model accommodates the essential features of multi-stable dynamic systems and provides an important framework for creating microwave devices and circuits.The main aim of this research work is to analyze the phenomenon of multi-stability and bifurcation in microwave engineering systems by applying Burger-Fisher equation. The study intends to obtain new solutions to the PDEs through the application of a new method, the exp (−Ψ(η))-expansion function method that uses rational, trigonometric, and hyperbolic traveling wave solutions. Existing research of the Burger-Fisher equation is not explicit and by solving the exact form the research aims at exploring the different conditions under which the equation behaves and studying the delicate interactions between the parameters of the multistable system.

Dynamical analysis and fault detection application of a time-delayed multi-stable stochastic resonance system driven by white correlated noises.

Bearing fault diagnosis is a crucial means to ensure the normal operation of machinery, reduce maintenance costs, enhance equipments safety and extend the service life of bearings. However, the noise interference in input signals often affects the effective extraction of bearing fault signals. In this paper, a time-delay multi-stable stochastic resonance (SR) system driven by white correlated noises is proposed. Firstly, the expressions of the steady-state probability density (SPD) function and the mean first passage time (MFPT) is derived. Simultaneously, we delve into the influences of various system parameters, including multiplicative noise intensity, additive noise intensity, noise correlated strength, time-delay, and time-delay feedback intensity, on the dynamical behavior exhibited by particles within the system. Then, according to the signal-to-noise ratio (SNR) formula, the paper investigates the impact of system parameters on the SNR. It is found that the ease of SR occurrence is directly related to the system parameters. Finally, the experimental results demonstrate that the proposed method exhibits superior performance in detecting faults compared with the classical bistable SR system and the quad-stable SR system.

Matryoshka multistability: Coexistence of an infinite number of exactly self-similar nested attractors in a fractal phase space

Multistability, and its special types such as megastability and extreme multistability, is an important phenomenon in modern nonlinear science that provides several possible practical applications. In this paper, we propose a new special type of multistability when the infinite number of exactly self-similar attractors nested inside each other coexist in a system. We called it matryoshka multistability due to its resemblance to the famous Russian wooden doll. We theoretically explain and experimentally confirm the properties of a new type of multistable behavior using two representative examples based on the Chua and Sprott Case J chaotic systems. In addition, we construct an adaptive controller for synchronizing two Chua-type matryoshka multistable systems when the amplitude of the master system is of arbitrary scale. The proposed type of multistability can find several applications in chaotic communication, cryptography, and data compression.

Learning noise-induced transitions by multi-scaling reservoir computing

Noise is usually regarded as adversarial to extracting effective dynamics from time series, such that conventional approaches usually aim at learning dynamics by mitigating the noisy effect. However, noise can have a functional role in driving transitions between stable states underlying many stochastic dynamics. We find that leveraging a machine learning model, reservoir computing, can learn noise-induced transitions. We propose a concise training protocol with a focus on a pivotal hyperparameter controlling the time scale. The approach is widely applicable, including a bistable system with white noise or colored noise, where it generates accurate statistics of transition time for white noise and specific transition time for colored noise. Instead, the conventional approaches such as SINDy and the recurrent neural network do not faithfully capture stochastic transitions even for the case of white noise. The present approach is also aware of asymmetry of the bistable potential, rotational dynamics caused by non-detailed balance, and transitions in multi-stable systems. For the experimental data of protein folding, it learns statistics of transition time between folded states, enabling us to characterize transition dynamics from a small dataset. The results portend the exploration of extending the prevailing approaches in learning dynamics from noisy time series.

Non-degenerate multi-stable discrete chaotic system for image encryption

Non-degenerate multi-stable discrete chaotic system for image encryption

Reprogrammable and reconfigurable mechanical computing metastructures with stable and high-density memory.

Mechanical computing encodes information in deformed states of mechanical systems, such as multistable structures. However, achieving stable mechanical memory in most multistable systems remains challenging and often limited to binary information. Here, we report leveraging coupling kinematic bifurcation in rigid cube-based mechanisms with elasticity to create transformable, multistable mechanical computing metastructures with stable, high-density mechanical memory. Simply stretching the planar metastructure forms a multistable corrugated platform. It allows for independent mechanical or magnetic actuation of individual bistable element, serving as pop-up voxels for display or binary units for various tasks such as information writing, erasing, reading, encryption, and mechanologic computing. Releasing the pre-stretched strain stabilizes the prescribed information, resistant to external mechanical or magnetic perturbations, whereas re-stretching enables editable mechanical memory, akin to selective zones or disk formatting for information erasure and rewriting. Moreover, the platform can be reprogrammed and transformed into a multilayer configuration to achieve high-density memory.

Statistical Mechanics Approaches for Studying Temperature and Rate Effects in Multistable Systems

Systems with a multistable energy landscape are widespread in physics, biophysics, technology, and materials science. They are strongly influenced by thermal fluctuations and external mechanical actions that can be applied at different rates, moving the system from equilibrium to non-equilibrium regimes. In this paper, we focus on a simple system involving a single breaking phenomenon to describe the various theoretical approaches used to study these problems. To begin with, we propose the exact solution at thermodynamic equilibrium based on the calculation of the partition function without approximations. We then introduce the technique of spin variables, which is able to simplify the treatment even for systems with a large number of coordinates. We then analyze the energy balance of the system to better understand its underlying physics. Finally, we introduce a technique based on transition state theory useful for studying the non-equilibrium dynamical regimes of these systems. This method is appropriate for the evaluation of rate effects and hysteresis loops. These approaches are developed for both the Helmholtz ensemble (prescribed extension) and the Gibbs ensemble (applied force) of statistical mechanics. The symmetry and duality of these two ensembles is discussed in depth. While these techniques are used here for a simple system with theoretical purposes, they can be applied to complex systems of interest for several physical, biophysical, and technological applications.

Basin entropy as an indicator of a bifurcation in a time-delayed system.

The basin entropy is a measure that quantifies, in a system that has two or more attractors, the predictability of a final state, as a function of the initial conditions. While the basin entropy has been demonstrated on a variety of multistable dynamical systems, to the best of our knowledge, it has not yet been tested in systems with a time delay, whose phase space is infinite dimensional because the initial conditions are functions defined in a time interval [-τ,0], where τ is the delay time. Here, we consider a simple time-delayed system consisting of a bistable system with a linear delayed feedback term. We show that the basin entropy captures relevant properties of the basins of attraction of the two coexisting attractors. Moreover, we show that the basin entropy can give an indication of the proximity of a Hopf bifurcation, but fails to capture the proximity of a pitchfork bifurcation. The Hopf bifurcation is detected because before the fixed points become unstable, a oscillatory, limit-cycle behavior appears that coexists with the fixed points. The new limit cycle modifies the structure of the basins of attraction, and this change is captured by basin entropy that reaches a maximum before the Hopf bifurcation. In contrast, the pitchfork bifurcation is not detected because the basins of attraction do not change as the bifurcation is approached. Our results suggest that the basin entropy can yield useful insights into the long-term predictability of time-delayed systems, which often have coexisting attractors.

Static vector solitons in a topological mechanical lattice

Topological solitons, renowned for their stability and particle-like collision behaviors, have sparked interest in developing macroscopic-scale information processing devices. However, the exploration of interactions between multiple topological solitons in mechanical systems remains elusive. In this study, we construct a topological mechanical lattice supporting static vector solitons that represent quantized degrees of freedom and can freely propagate across the system. Drawing inspiration from coupled double atomic chains with sublattice symmetry breaking, we design a mechanical analogue featuring topologically protected boundary modes and induce independent modes to finite motions along branched motion pathways. Through a continuum theory, we describe the evolution of boundary modes with vector solitons composed of superposed kink solutions, identifying them as minimum energy pathways on the rugged effective potential surface with multiple degenerate ground states. Our results reveal the connection between transformable topological lattices and multistable systems, providing insight into nonlinear topological mechanics.

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.

Harnessing multi-stable piezoelectric systems for enhanced wind energy harvesting

With the ongoing evolution of microelectronic devices toward lower power consumption, the utilization of piezoelectric materials for energy harvesting from wind-induced vibrations has garnered considerable attention. This study employs a combined approach involving finite element analysis and experiments to investigate the energy harvesting efficiency of the multi-stable piezoelectric wind energy harvester (MPWEH) and compares its performance with two alternative systems. The MPWEH demonstrates higher strains in both the x and y directions during reciprocating cross-well vibrations, establishing its superior energy harvesting efficiency compared to the alternative systems. Notably, at a wind speed of 8 m s−1, the MPWEH generates an output power nearly six times higher than local bistable piezoelectric energy harvester (LBPEH). The MPWEH achieves the maximum power density of 9.8125 mW cm−3, whereas the LBPEH registers the power density of 1.625 mW cm−3. The experimental results indicate that, under the optimal load resistance of 40 kΩ and a wind speed of 14 m s−1, the MPWEH achieves a peak output power of 2.76 mW, with a power density of 17.25 mW cm−3. The versatile applicability of the MPWEH extends across various low-power consumption microelectronic devices, positioning it as a valuable candidate for empowering continuous monitoring sensors in diverse domains.

Tuning the Properties of Multi-Stable Structures Post-Fabrication Via the Two-Way Shape Memory Polymer Effect.

Multi-stable elements are commonly employed to design reconfigurable and adaptive structures, because they enable large and reversible shape changes in response to changing loads, while simultaneously allowing self-locking capabilities. However, existing multi-stable structures have properties that depend on their initial design and cannot be tailored post-fabrication. Here, a novel design approach is presented that combines multi-stable structures with two-way shape memory polymers. By leveraging both the one-way and two-way shape memory effect under bi-axial strain conditions, the structures can re-program their 3D shape, bear loads, and self-actuate. Results demonstrate that the structures' shape and stiffness can be tuned post-fabrication at the user's need and the multi-stability can be suppressed or activated on command. The control of multi-stability prevents undesired snapping of the structures and enables higher load-bearing capability, compared to conventional multi-stable systems. The proposed approach offers the possibility to augment the functionality of existing multi-stable concepts, showing potential for the realization of highly adaptable mechanical structures that can reversibly switch between being mono and multi-stable and that can undergo shape changes in response to a change in temperature.

Experimental switching between coexisting attractors in the yoke-bell-clapper system.

This paper presents experimental switching between two attractors in the swinging bell. In the considered yoke-bell-clapper system, two coexisting solutions appear. In the first one, we observe a single impact between the bell and the clapper per one period of motion, and in the second solution, no impacts occur-no sound is produced. Based on the time-dependent stability margin method, we numerically detect parts of the trajectories where the system is most prone to perturbations. Using this knowledge, we experimentally investigate switching between attractors by applying the perturbation to the clapper. We show that we can easily enforce the change of attractor by properly timing the perturbation. The results prove that, based on the results from the time-dependent stability margin numerical method, we are able to effectively alter the wrong operation of the bell (lack of impact) to the correct operation (solution with impact). The analysis is conducted on the real-world mechanical system rather than paradigmatic examples. Therefore, it contributes to the subject of multistability and nonlinearity in engineering design. Novel, recently developed methods for analyzing multistable systems are successfully employed during the investigation. The paper shows that a complex phenomenon of multistability observed in the system, which is considered simple and undemanding from an engineering design point of view.

Multistable States and Snap-Through Instabilities in an Interconnected Dielectric Elastomer Actuators System

Abstract Snap-through instabilities in single and interconnected dielectric elastomer actuators have demonstrated their potential in enabling many functionalities such as large deformation, high-speed actuation, and enhanced flowrates in fluid pumps. However, the nonlinear nature of dielectric elastomers and the complex interplay of mechanics in interconnected inflated actuators, make the modeling of such systems challenging. In this paper, we present a methodology to analytically model the instabilities in a system of three interconnected homogeneous spherical dielectric elastomer actuators through graphical and numerical approaches. The simulation results reveal the presence of a locus of initial stable states that the interconnected actuators can achieve at zero voltage. In specific loading conditions, the system exhibits multiple stable states which can be cyclically transitioned between by selectively applying voltage to specific actuators. In other conditions, the system may undergo two successive instabilities when the voltage applied to a single actuator in the system is increased monotonically. These results retrieve existing experimental results theoretically for the first time and identify a new behavior of cascading instabilities in inflated dielectric elastomer actuators. We hope this work will pave the way for programmable design of multistable systems to unravel new capabilities in dielectric elastomer actuators and soft robotics.

A novel multistable chaotic system with 2m-scroll attractor and its application

A novel multistable chaotic system with 2m-scroll attractor and its application

Synchronization in network motifs of three piecewise Rössler systems coupled in ring configuration

In this work, we study the synchronization in the network motifs of three piecewise Rössler systems that exhibit coexistence of two attractors, coupled in ring configuration, in particular a bidirectional auxiliary coupling. For this configuration, we analyze the interaction of the systems by varying the coupling strengths, in order to characterize the conditions under which complete synchronization between the three systems can be achieved. The numerical results prove that the interaction of various multistable chaotic systems causes different types of synchronization and very complex dynamics between the systems is achieved. We obtain complete synchronization between the three systems and find the range of values of the coupling strengths where the distinct types of synchronization occur.

Heterogeneous multistability and antimonotonicity for a new 3D system with a triple well nonlinearity: theoretical study, control and microcontroller implementation

We propose a new 3D autonomous multistable jerk-like system with a nonlinear term consisting of a six-order triple well function. The presence of six equilibrium points with symmetrical locations along the x-axis represents one of the main distinguishing properties of the new system. Strikingly, the stability analysis of equilibria reveals a cascade of Hopf bifurcations at three specific values of a single control parameter, which results in several forms of complexity. Accordingly, various forms of coexisting attractors such as stable fixed points, limit cycles of diverse periodicities, and chaotic attractors are depicted for some special parameter values. Moreover, It is found that the new jerk-like system with six order triple well polynomial function exhibit extremely complex nonlinear behaviors such as anti-monotone bifurcations, hysteresis and parallel bifurcation branches. These latter aspects explain the presence of multiple (i.e. up to four) coexisting asymmetric attractors for some special rank of parameters. In the presence of multiple competing dynamics, we resort to basins of attraction in order to highlight the how the state space is magnetized. The combination of dynamic features discussed in the new jerk-like system with triple well polynomials nonlinearity introduced in this article is unique and rarely reported. An electronic version of the new system with triple well polynomial nonlinearity is implemented in PSpice. Moreover, a hardware digital implementation of the system is also carried out using an Arduino microcontroller. A very good agreement is captured between PSpice simulation results, the laboratory measurements and the theoretical predictions.

Multi-stable hidden attractor chaotic system and its analog coexistence circuit realization

This paper proposes a multi-stable chaotic system with relatively complex hidden attractors. The dynamic performance of chaotic systems is under investigation via numerical simulations such as Lyapunov exponents, division diagrams, and phase diagrams, and it has been further found that the chaotic system with hidden attractors can switch between the two cases of having no equilibrium or having two stable equilibria. Due to the system’s symmetry, multiple co-existing attractors can be obtained by choosing appropriate parameters and initial values. It is demonstrated that the system exhibits a multi-stability phenomenon, which means that different initial conditions generate two or more different dynamics. In addition, the variational method is used to explore short-period orbits of a topological length up to 3, which provides a deeper understanding of the essential components of a chaotic system. Finally, circuit implementation verifies its correctness and feasibility.

A novel four-wing chaotic system with multiple equilibriums: Dynamical analysis, multistability, circuit simulation and pseudo random number generator (PRNG) based on the voice encryption

Recently, there has been tremendous interest worldwide in the possibility of using chaos in communication systems. Many different chaos-based secure communication schemes have been proposed up until now. However, systems with strong chaoticity are more suitable for chaos-based secure communication. From the viewpoint of Lyapunov exponents, a chaotic system with a larger positive Lyapunov exponent is said to be more complex. This paper constructing a multistable chaotic system that can produce coexisting attractors is an attractive field of research due to its theoretical and practical usefulness. An innovative 3D dynamical system is presented in this research. It can display various coexisting attractors for the same values of parameters. The new system is more suitable for chaos-based applications than recently reported systems since it exhibits strong multistable chaotic behavior, as proved by its large positive Lyapunov exponent. Furthermore, the accuracy of the numerical calculation and the system's physical implementations are confirmed by analog circuit simulation. Finally, implementing the proposed voice encryption is done using a four-wing chaotic system based on the PRNG.

A New Memristive System with Extreme Multistability and Hidden Chaotic Attractors and with Application to Image Encryption

Chaotic systems have proven highly beneficial in engineering applications. Pseudo-random numbers produced by chaotic systems have been used for secure communication, notably image encryption. Specific characteristics can increase the chaotic behavior of the system by adding complexity and nonlinearity. The three most well-known characteristics are memristive properties, multistability (coexisting attractors), and hidden attractors. These characteristics strengthen the produced time series’ unpredictability and randomness, strengthening an encryption algorithm’s resistance to many attacks. This study introduces a unique four-dimensional chaotic system with extreme multistability with respect to three initial conditions (including the memristor initial condition) and all previously known properties. It is rare to find an extreme multistable system like this. This system is coupled with a quadratic flux-controlled memristor based on the well-known Sprott J system. This system has a line of unstable equilibrium points with hidden attractors. The memristor displays the characteristic pinched hysteresis loops, where the area inside a loop and the voltage frequency are inversely related. A comprehensive dynamical analysis thoroughly examines all system characteristics and initial conditions. The numerical findings are carefully verified, and an analog circuit is successfully built and simulated. The chaotic sequences generated by this system are combined with deoxyribonucleic acid (DNA) operations and the global bit scrambling (GBS) technique to create an image encryption algorithm that has strong resistance to a variety of potential attacks, including noise, statistical, exhaustive, differential, and cropping attacks.