Experimental observation of extreme multistability in an electronic system of two coupled Rössler oscillators

Memristor-coupled dual-neuron mapping model: initials-induced coexisting firing patterns and synchronization activities

Synaptic plasticity makes memristors particularly suitable for simulating the connection synapses between neurons that describe magnetic induction coupling. By applying a memristor to the synaptic coupling between two map-based neuron models, a memristor-coupled dual-neuron mapping (MCDN) model is proposed in this article. The MCDN model has a line fixed point set associated with the memristor initial state, which is always unstable for the model parameters and memristor initial state of interest. Complex spiking/bursting firing patterns and their transitions are disclosed using some dynamical analysis means. The numerical results show that these spiking/bursting firings are significantly relied on the memristor initial state, demonstrating the coexistence of firing patterns. Moreover, the initial effects of complete synchronization are explored for the homogeneous MCDN model, and it is clarified that in addition to being related to the coupling strength, the synchronization activities are extremely dependent on the initial states of the memristor and neurons. Finally, these numerical results are confirmed by the FPGA-based hardware experiments.

On multistability near the boundary of generalized synchronization in unidirectionally coupled chaotic systems

Multistability in the intermittent generalized synchronization regime in unidirectionally coupled chaotic systems has been found. To study such a phenomenon, the method for revealing the existence of multistable states in interacting systems being the modification of an auxiliary system approach has been proposed. The efficiency of the method has been testified using the examples of unidirectionally coupled logistic maps and Rössler systems being in the intermittent generalized synchronization regime. The quantitative characteristic of multistability has been introduced into consideration.

Bistability and hidden attractors in the paradigmatic Rössler'76 system

In this paper, the dynamics of the paradigmatic Rössler system is investigated in a yet unexplored region of its three-dimensional parameter space. We prove a necessary condition in this space for which the Rössler system can be chaotic. By using standard numerical tools, like bifurcation diagrams, Poincaré sections, and first-return maps, we highlight both asymptotically stable limit cycles and chaotic attractors. Lyapunov exponents are used to verify the chaotic behavior while random numerical procedures and various plane cross sections of the basins of attraction of the coexisting attractors prove that both limit cycles and chaotic attractors are hidden. We thus obtain previously unknown examples of bistability in the Rössler system, where a point attractor coexists with either a hidden limit cycle attractor or a hidden chaotic attractor.

Synchronization within synchronization: transients and intermittency in ecological networks

Transients are fundamental to ecological systems with significant implications to management, conservation and biological control. We uncover a type of transient synchronization behavior in spatial ecological networks whose local dynamics are of the chaotic, predator–prey type. In the parameter regime where there is phase synchronization among all the patches, complete synchronization (i.e. synchronization in both phase and amplitude) can arise in certain pairs of patches as determined by the network symmetry—henceforth the phenomenon of  ‘synchronization within synchronization.’ Distinct patterns of complete synchronization coexist but, due to intrinsic instability or noise, each pattern is a transient and there is random, intermittent switching among the patterns in the course of time evolution. The probability distribution of the transient time is found to follow an algebraic scaling law with a divergent average transient lifetime. Based on symmetry considerations, we develop a stability analysis to understand these phenomena. The general principle of symmetry can also be exploited to explain previously discovered, counterintuitive synchronization behaviors in ecological networks.

Noise-enabled species recovery in the aftermath of a tipping point

The beneficial role of noise in promoting species coexistence and preventing extinction has been recognized in theoretical ecology, but previous studies were mostly concerned with low-dimensional systems. We investigate the interplay between noise and nonlinear dynamics in real-world complex mutualistic networks with a focus on species recovery in the aftermath of a tipping point. Particularly, as a critical parameter such as the mutualistic interaction strength passes through a tipping point, the system collapses and approaches an extinction state through a dramatic reduction in the species populations to near-zero values. We demonstrate the striking effect of noise: when the direction of parameter change is reversed through the tipping point, noise enables species recovery which otherwise would not be possible. We uncover an algebraic scaling law between the noise amplitude and the parameter distance from the tipping point to the recovery point and provide a physical understanding through analyzing the nonlinear dynamics based on an effective, reduced-dimension model. Noise, in the form of small population fluctuations, can thus play a positive role in protecting high-dimensional, complex ecological networks.

Multistability in a physical memristor-based modified Chua's circuit

A physical Sr_0.95Ba_0.05TiO₃ memristor-based modified Chua's circuit is proposed, which is studied in this paper by means of both theoretical analysis and numerical simulations. The stability of this memristor-based modified Chua's circuit is analyzed. A systematic investigation of the dynamic behaviors and their dependence on the initial states and circuit parameters is performed, presenting the Lyapunov exponents spectra, bifurcation diagrams, phase diagrams, and Poincaré maps. The circuit emerges multiple dynamic behaviors, including stable points, periodic cycles, chaos, and some transient chaos. Specifically, the system has multistability, i.e., coexisting periods and coexisting chaotic attractors with the change of initial states and circuit parameters. This study is conducive to the subsequent design and analysis of memristor-based circuits for potential practical applications.

On multistability behavior of unstable dissipative systems

We present dissipative systems with unstable dynamics called the unstable dissipative systems which are capable of generating a multi-stable behavior, i.e., depending on its initial condition, the trajectory of the system converges to a specific attractor. Piecewise linear (PWL) systems are generated based on unstable dissipative systems, whose main attribute when they are switched is the generation of chaotic trajectories with multiple wings or scrolls. For this PWL system, a structure is proposed where both the linear part and the switching function depend on two parameters. We show the range of values of such parameters where the PWL system presents a multistable behavior and trajectories with multiscrolls.

Emergence, evolution, and control of multistability in a hybrid topological quantum/classical system

We present a novel class of nonlinear dynamical systems-a hybrid of relativistic quantum and classical systems and demonstrate that multistability is ubiquitous. A representative setting is coupled systems of a topological insulator and an insulating ferromagnet, where the former possesses an insulating bulk with topologically protected, dissipationless, and conducting surface electronic states governed by the relativistic quantum Dirac Hamiltonian and the latter is described by the nonlinear classical evolution of its magnetization vector. The interactions between the two are essentially the spin transfer torque from the topological insulator to the ferromagnet and the local proximity induced exchange coupling in the opposite direction. The hybrid system exhibits a rich variety of nonlinear dynamical phenomena besides multistability such as bifurcations, chaos, and phase synchronization. The degree of multistability can be controlled by an external voltage. In the case of two coexisting states, the system is effectively binary, opening a door to exploitation for developing spintronic memory devices. Because of the dissipationless and spin-momentum locking nature of the surface currents of the topological insulator, little power is needed for generating a significant current, making the system appealing for potential applications in next generation of low power memory devices.

Multistability in Chua's circuit with two stable node-foci

Only using one-stage op-amp based negative impedance converter realization, a simplified Chua's diode with positive outer segment slope is introduced, based on which an improved Chua's circuit realization with more simpler circuit structure is designed. The improved Chua's circuit has identical mathematical model but completely different nonlinearity to the classical Chua's circuit, from which multiple attractors including coexisting point attractors, limit cycle, double-scroll chaotic attractor, or coexisting chaotic spiral attractors are numerically simulated and experimentally captured. Furthermore, with dimensionless Chua's equations, the dynamical properties of the Chua's system are studied including equilibrium and stability, phase portrait, bifurcation diagram, Lyapunov exponent spectrum, and attraction basin. The results indicate that the system has two symmetric stable nonzero node-foci in global adjusting parameter regions and exhibits the unusual and striking dynamical behavior of multiple attractors with multistability.

Extreme multistability: Attractor manipulation and robustness

The coexistence of infinitely many attractors is called extreme multistability in dynamical systems. In coupled systems, this phenomenon is closely related to partial synchrony and characterized by the emergence of a conserved quantity. We propose a general design of coupling that leads to partial synchronization, which may be a partial complete synchronization or partial antisynchronization and even a mixed state of complete synchronization and antisynchronization in two coupled systems and, thereby reveal the emergence of extreme multistability. The proposed design of coupling has wider options and allows amplification or attenuation of the amplitude of the attractors whenever it is necessary. We demonstrate that this phenomenon is robust to parameter mismatch of the coupled oscillators.

Organization and identification of solutions in the time-delayed Mackey-Glass model

Multistability in the long term dynamics of the Mackey-Glass (MG) delayed model is analyzed by using an electronic circuit capable of controlling the initial conditions. The system's phase-space is explored by varying the parameter values of two families of initial functions. The evolution equation of the electronic circuit is derived and it is shown that, in the continuous limit, it exactly corresponds to the MG model. In practice, when using a finite set of capacitors, an excellent agreement between the experimental observations and the numerical simulations is manifested. As the delay is increased, different periodic or aperiodic solutions appear. We observe abundant periodic solutions that have the same period but a different alternation of peaks of dissimilar amplitudes and propose a novel symbolic method to classify these solutions.

Synchronization of intermittent behavior in ensembles of multistable dynamical systems

We propose a methodology to analyze synchronization in an ensemble of diffusively coupled multistable systems. First, we study how two bidirectionally coupled multistable oscillators synchronize and demonstrate the high complexity of the basins of attraction of coexisting synchronous states. Then, we propose the use of the master stability function (MSF) for multistable systems to describe synchronizability, even during intermittent behavior, of a network of multistable oscillators, regardless of both the number of coupled oscillators and the interaction structure. In particular, we show that a network of multistable elements is synchronizable for a given range of topology spectra and coupling strengths, irrespective of specific attractor dynamics to which different oscillators are locked, and even in the presence of intermittency. Finally, we experimentally demonstrate the feasibility and robustness of the MSF approach with a network of multistable electronic circuits.

Reply to "comment on 'how to obtain extreme multistability in coupled dynamical systems' "

In this Reply we answer the two major issues raised by the Comment. First, we point out that the idea of constructing extreme multistability in simple dynamical systems is not new and has been demonstrated previously by other authors. Furthermore, we emphasize the importance of the concept of a conserved quantity and its consequences for the dynamics, which applies to all the examples in the Comment. Second, we show that the design of controllers to achieve extreme multistability in coupled systems is as general as described in Phys. Rev. E 85, 035202(R) (2012) by providing two examples which do not lead to a master-slave dynamics.