Why are chaotic attractors rare in multistable systems?

Control, bi-stability, and preference for chaos in time-dependent vaccination campaign

In this work, effects of constant and time-dependent vaccination rates on the Susceptible-Exposed-Infected-Recovered-Susceptible (SEIRS) seasonal model are studied. Computing the Lyapunov exponent, we show that typical complex structures, such as shrimps, emerge for given combinations of a constant vaccination rate and another model parameter. In some specific cases, the constant vaccination does not act as a chaotic suppressor and chaotic bands can exist for high levels of vaccination (e.g., >0.95). Moreover, we obtain linear and non-linear relationships between one control parameter and constant vaccination to establish a disease-free solution. We also verify that the total infected number does not change whether the dynamics is chaotic or periodic. The introduction of a time-dependent vaccine is made by the inclusion of a periodic function with a defined amplitude and frequency. For this case, we investigate the effects of different amplitudes and frequencies on chaotic attractors, yielding low, medium, and high seasonality degrees of contacts. Depending on the parameters of the time-dependent vaccination function, chaotic structures can be controlled and become periodic structures. For a given set of parameters, these structures are accessed mostly via crisis and, in some cases, via period-doubling. After that, we investigate how the time-dependent vaccine acts in bi-stable dynamics when chaotic and periodic attractors coexist. We identify that this kind of vaccination acts as a control by destroying almost all the periodic basins. We explain this by the fact that chaotic attractors exhibit more desirable characteristics for epidemics than periodic ones in a bi-stable state.

Mechanism of multistability in chaotic maps

This research aims to investigate the mechanisms of multistability in chaotic maps. The study commences by examining the fundamental principles governing the development of homogeneous multistability using a basic one-dimensional chain-climbing map. Findings suggest that the phase space can be segmented into distinct uniform mediums where particles exhibit consistent movement. As critical parameter values are reached, channels emerge between these mediums, resulting in deterministic chaotic diffusion. Additionally, the study delves into the topic of introducing heterogeneous factors on the formation of heterogeneous multistability in the one-dimensional map. A thorough examination of phenomena such as multistate intermittency highlights the intimate connection between specific phase transition occurrences and channel formation. Finally, by analyzing two instances-a memristive chaotic map and a hyperchaotic map-the underlying factors contributing to the emergence of multistability are scrutinized. This study offers an alternative perspective for verifying the fundamental principles of homogenous and heterogeneous multistability in complex high-dimensional chaotic maps.

Dynamics, multistability, and crisis analysis of a sine-circle nontwist map

We propose a one-dimensional dynamical system, the sine-circle nontwist map, that can be considered a local approximation of the standard nontwist map and an extension of the paradigmatic sine-circle map. The map depends on three parameters, exhibiting a simple mathematical form but with a rich dynamical behavior. We identify periodic, quasiperiodic, and chaotic solutions for different parameter sets with the Lyapunov exponent and Slater's theorem. From the bifurcation analysis, we determine two bifurcation lines, those that depend on just two of the control parameters, for which the bifurcation that occurs is of the saddle-node type. In order to investigate multistability, we analyze the bifurcation diagrams in the two directions of parameter variation and we observe some regions of hysteresis, representing the coexistence of different attractors. We also analyze different multistable scenarios, as single attractor, coexistence of periodic attractors, coexistence of chaotic and periodic attractors, chaotic behavior, and coexistence of different chaotic bands, by the Lyapunov exponent and the analysis of the domain occupied by the solutions. From the parameter spaces constructed, we observe the prevalence of single attractor and only chaotic behavior scenarios. The multistable scenario is, mostly, formed by different periodic attractors. Lastly, we analyze the crisis in chaotic attractors and we identify the interior and the boundary crisis. From our results, the boundary crisis plays a key role for the extinction of multistability.

Control of tipping points in stochastic mutualistic complex networks

Nonlinear stochastic complex networks in ecological systems can exhibit tipping points. They can signify extinction from a survival state and, conversely, a recovery transition from extinction to survival. We investigate a control method that delays the extinction and advances the recovery by controlling the decay rate of pollinators of diverse rankings in a pollinators-plants stochastic mutualistic complex network. Our investigation is grounded on empirical networks occurring in natural habitats. We also address how the control method is affected by both environmental and demographic noises. By comparing the empirical network with the random and scale-free networks, we also study the influence of the topological structure on the control effect. Finally, we carry out a theoretical analysis using a reduced dimensional model. A remarkable result of this work is that the introduction of pollinator species in the habitat, which is immune to environmental deterioration and that is in mutualistic relationship with the collapsed ones, definitely helps in promoting the recovery. This has implications for managing ecological systems.

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.

Role of noise and parametric variation in the dynamics of gene regulatory circuits

Stochasticity in gene expression impacts the dynamics and functions of gene regulatory circuits. Intrinsic noises, including those that are caused by low copy number of molecules and transcriptional bursting, are usually studied by stochastic simulations. However, the role of extrinsic factors, such as cell-to-cell variability and heterogeneity in the microenvironment, is still elusive. To evaluate the effects of both the intrinsic and extrinsic noises, we develop a method, named sRACIPE, by integrating stochastic analysis with random circuit perturbation (RACIPE) method. RACIPE uniquely generates and analyzes an ensemble of models with random kinetic parameters. Previously, we have shown that the gene expression from random models form robust and functionally related clusters. In sRACIPE we further develop two stochastic simulation schemes, aiming to reduce the computational cost without sacrificing the convergence of statistics. One scheme uses constant noise to capture the basins of attraction, and the other one uses simulated annealing to detect the stability of states. By testing the methods on several synthetic gene regulatory circuits and an epithelial-mesenchymal transition network in squamous cell carcinoma, we demonstrate that sRACIPE can interpret the experimental observations from single-cell gene expression data. We observe that parametric variation (the spread of parameters around a median value) increases the spread of the gene expression clusters, whereas high noise merges the states. Our approach quantifies the robustness of a gene circuit in the presence of noise and sheds light on a new mechanism of noise-induced hybrid states. We have implemented sRACIPE as an R package.

Multistability in the cyclic competition system

Cyclically competition models have been successful to gain an insight of biodiversity mechanism in ecosystems. There are, however, still limitations to elucidate complex phenomena arising in real competition. In this paper, we report that a multistability occurs in a simple rock-paper-scissor cyclically competition model by assuming that intraspecific competition depends on the logistic growth of each species density. This complex stability is absent in any cyclically competition model, and we investigate how the proposed intraspecific competition affects biodiversity in the existing society of three species through macroscopic and microscopic approaches. When the system is multistable, we show basins of the asymptotically stable heteroclinic cycle and stable attractors to demonstrate how the survival state is determined by initial densities of three species. Also, we find that the multistability is associated with a subcritical Hopf bifurcation. This surprising finding will give an opportunity to interpret rich dynamical phenomena in ecosystems which may occur in cyclic competition systems with different types of interactions.

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.

Survivability of Deterministic Dynamical Systems

The notion of a part of phase space containing desired (or allowed) states of a dynamical system is important in a wide range of complex systems research. It has been called the safe operating space, the viability kernel or the sunny region. In this paper we define the notion of survivability: Given a random initial condition, what is the likelihood that the transient behaviour of a deterministic system does not leave a region of desirable states. We demonstrate the utility of this novel stability measure by considering models from climate science, neuronal networks and power grids. We also show that a semi-analytic lower bound for the survivability of linear systems allows a numerically very efficient survivability analysis in realistic models of power grids. Our numerical and semi-analytic work underlines that the type of stability measured by survivability is not captured by common asymptotic stability measures.

Multistability, chaos, and random signal generation in semiconductor superlattices

Historically, semiconductor superlattices, artificial periodic structures of different semiconductor materials, were invented with the purpose of engineering or manipulating the electronic properties of semiconductor devices. A key application lies in generating radiation sources, amplifiers, and detectors in the "unusual" spectral range of subterahertz and terahertz (0.1-10 THz), which cannot be readily realized using conventional radiation sources, the so-called THz gap. Efforts in the past three decades have demonstrated various nonlinear dynamical behaviors including chaos, suggesting the potential to exploit chaos in semiconductor superlattices as random signal sources (e.g., random number generators) in the THz frequency range. We consider a realistic model of hot electrons in semiconductor superlattice, taking into account the induced space charge field. Through a systematic exploration of the phase space we find that, when the system is subject to an external electrical driving of a single frequency, chaos is typically associated with the occurrence of multistability. That is, for a given parameter setting, while there are initial conditions that lead to chaotic trajectories, simultaneously there are other initial conditions that lead to regular motions. Transition to multistability, i.e., the emergence of multistability with chaos as a system parameter passes through a critical point, is found and argued to be abrupt. Multistability thus presents an obstacle to utilizing the superlattice system as a reliable and robust random signal source. However, we demonstrate that, when an additional driving field of incommensurate frequency is applied, multistability can be eliminated, with chaos representing the only possible asymptotic behavior of the system. In such a case, a random initial condition will lead to a trajectory landing in a chaotic attractor with probability 1, making quasiperiodically driven semiconductor superlattices potentially as a reliable device for random signal generation to fill the THz gap. The interplay among noise, multistability, and chaos is also investigated.

A geometrical approach to control and controllability of nonlinear dynamical networks

In spite of the recent interest and advances in linear controllability of complex networks, controlling nonlinear network dynamics remains an outstanding problem. Here we develop an experimentally feasible control framework for nonlinear dynamical networks that exhibit multistability. The control objective is to apply parameter perturbation to drive the system from one attractor to another, assuming that the former is undesired and the latter is desired. To make our framework practically meaningful, we consider restricted parameter perturbation by imposing two constraints: it must be experimentally realizable and applied only temporarily. We introduce the concept of attractor network, which allows us to formulate a quantifiable controllability framework for nonlinear dynamical networks: a network is more controllable if the attractor network is more strongly connected. We test our control framework using examples from various models of experimental gene regulatory networks and demonstrate the beneficial role of noise in facilitating control.

Nonlinear dynamics induced anomalous Hall effect in topological insulators

We uncover an alternative mechanism for anomalous Hall effect. In particular, we investigate the magnetisation dynamics of an insulating ferromagnet (FM) deposited on the surface of a three-dimensional topological insulator (TI), subject to an external voltage. The spin-polarised current on the TI surface induces a spin-transfer torque on the magnetisation of the top FM while its dynamics can change the transmission probability of the surface electrons through the exchange coupling and hence the current. We find a host of nonlinear dynamical behaviors including multistability, chaos, and phase synchronisation. Strikingly, a dynamics mediated Hall-like current can arise, which exhibits a nontrivial dependence on the channel conductance. We develop a physical understanding of the mechanism that leads to the anomalous Hall effect. The nonlinear dynamical origin of the effect stipulates that a rich variety of final states exist, implying that the associated Hall current can be controlled to yield desirable behaviors. The phenomenon can find applications in Dirac-material based spintronics.

Detection meeting control: Unstable steady states in high-dimensional nonlinear dynamical systems

We articulate an adaptive and reference-free framework based on the principle of random switching to detect and control unstable steady states in high-dimensional nonlinear dynamical systems, without requiring any a priori information about the system or about the target steady state. Starting from an arbitrary initial condition, a proper control signal finds the nearest unstable steady state adaptively and drives the system to it in finite time, regardless of the type of the steady state. We develop a mathematical analysis based on fast-slow manifold separation and Markov chain theory to validate the framework. Numerical demonstration of the control and detection principle using both classic chaotic systems and models of biological and physical significance is provided.

Different types of critical behavior in conservatively coupled Hénon maps

We study the dynamics of two conservatively coupled Hénon maps at different levels of dissipation. It is shown that the decrease of dissipation leads to changes in the structure of the parameter plane and the scenarios of transition to chaos compared to the case of infinitely strong dissipation. Particularly, the Feigenbaum line becomes divided into several fragments. Some of these fragments have critical points of different types, namely, of C and H type, as their terminal points. Also the mechanisms of formation of these Feigenbaum line ruptures are described.

Complex dynamics in nanosystems

Complex dynamics associated with multistability have been studied extensively in the past but mostly for low-dimensional nonlinear dynamical systems. A question of fundamental interest is whether multistability can arise in high-dimensional physical systems. Motivated by the ever increasing widespread use of nanoscale systems, we investigate a prototypical class of nanoelectromechanical systems: electrostatically driven Si nanowires, mathematically described by a set of driven, nonlinear partial differential equations. We develop a computationally efficient algorithm to solve the equations. Our finding is that multistability and complicated structures of basins of attraction are common types of dynamics, and the latter can be attributed to extensive transient chaos. Implications of these phenomena to device operations are discussed.

Noisy scattering dynamics in the randomly driven Hénon-Heiles oscillator

Noisy scattering dynamics in the randomly driven Hénon-Heiles oscillator is investigated when the energy is above the threshold to permit particles to escape from the scattering region. First, some basic simulation procedures are briefly introduced and the fractal exit basins appear to be robust when the bounded noisy excitation is imposed on the oscillator. Second, several key fractal characteristics of the sample basin boundaries, such as the delay-time function and the uncertainty dimension, are estimated from which this oscillator is found to be structurally unstable against the bounded noisy excitation. Moreover, the stable and unstable manifolds of some sample chaotic invariant sets are estimated and illustrated in a special two-dimensional Poincaré section. Lastly, several previous methods are developed to identify three arbitrarily chosen noisy scattering time series of the randomly driven Hénon-Heiles oscillator, from which the quasiperiodic-dominant and the chaotic-dominant dynamical behaviors are distinguished.

Emerging attractors and the transition from dissipative to conservative dynamics

The topological structure of basin boundaries plays a fundamental role in the sensitivity to the final state in chaotic dynamical systems. Herewith we present a study on the dynamics of dissipative systems close to the Hamiltonian limit, emphasizing the increasing number of periodic attractors, and on the structural changes in their basin boundaries as the dissipation approaches zero. We show numerically that a power law with nontrivial exponent describes the growth of the total number of periodic attractors as the damping is decreased. We also establish that for small scales the dynamics is governed by effective dynamical invariants, whose measure depends not only on the region of the phase space but also on the scale under consideration. Therefore, our results show that the concept of effective invariants is also relevant for dissipative systems.

Constructive analysis of noise-induced transitions for coexisting periodic attractors of the Lorenz model

We study the stochastically forced Lorenz model in the parameter zone admitting two coexisting limit cycles under the transition to chaos via period-doubling bifurcations. Noise-induced transitions between both different parts of the single attractor and two coexisting separate attractors are demonstrated. The effects of structural stabilization and noise symmetrization are discussed. We suggest a stochastic sensitivity function technique for the analysis of noise-induced transitions between two coexisting limit cycles. This approach allows us to construct the dispersion ellipses of random trajectories for any Poincare sections. Possibilities of our descriptive-geometric method for a detailed analysis of noise-induced transitions between two periodic attractors of Lorenz model are demonstrated.

Multistability and arithmetically period-adding bifurcations in piecewise smooth dynamical systems

Multistability has been a phenomenon of continuous interest in nonlinear dynamics. Most existing works so far have focused on smooth dynamical systems. Motivated by the fact that nonsmooth dynamical systems can arise commonly in realistic physical and engineering applications such as impact oscillators and switching electronic circuits, we investigate multistability in such systems. In particular, we consider a generic class of piecewise smooth dynamical systems expressed in normal form but representative of nonsmooth systems in realistic situations, and focus on the weakly dissipative regime and the Hamiltonian limit. We find that, as the Hamiltonian limit is approached, periodic attractors can be generated through a series of saddle-node bifurcations. A striking phenomenon is that the periods of the newly created attractors follow an arithmetic sequence. This has no counterpart in smooth dynamical systems. We provide physical analyses, numerical computations, and rigorous mathematical arguments to substantiate the finding.

Dissipation as a mechanism of energy gain

Some properties of the annular billiard under the presence of weak dissipation are studied. We show, in a dissipative system, that the average energy of a particle acquires higher values than its average energy of the conservative case. The creation of attractors, associated with a chaotic dynamics in the conservative regime, both in appropriated regions of the phase space, constitute a generic mechanism to increase the average energy of dynamical systems.

Fractal dimension in dissipative chaotic scattering

The effect of weak dissipation on chaotic scattering is relevant to situations of physical interest. We investigate how the fractal dimension of the set of singularities in a scattering function varies as the system becomes progressively more dissipative. A crossover phenomenon is uncovered where the dimension decreases relatively more rapidly as a dissipation parameter is increased from zero and then exhibits a much slower rate of decrease. We provide a heuristic theory and numerical support from both discrete-time and continuous-time scattering systems to establish the generality of this phenomenon. Our result is expected to be important for physical phenomena such as the advection of inertial particles in open chaotic flows, among others.

Basin bifurcations in quasiperiodically forced coupled systems

We study the effect of quasiperiodic forcing on a system of coupled identical logistic maps. Upon a variation of system parameters, a variety of different dynamical regimes can be observed, including phenomena such as bistability and multistability. At the bifurcation to bistability, in a manner reminiscent of attractor expansion at interior crises, there is an abrupt change in the size of attractor basins. In the bistable region, attractor basins undergo additional bifurcations wherein holes and islands are created within the basins when system parameters change. These can be understood by examining critical surfaces for the coupled system.

Basin size evolution between dissipative and conservative limits

Recent methods for stabilizing systems like, e.g., loss-modulated CO2 lasers, involve inducing controlled monostability via slow parameter modulations. However, such stabilization methods presuppose detailed knowledge of the structure and size of basins of attraction. In this Brief Report, we numerically investigate basin size evolution when parameters are varied between dissipative and conservative limits. Basin volumes shrink fast as the conservative limit is approached, being well approximated by Gaussian profiles, independently of the period. Basin shrinkage and vanishing is due to the absence of bounded motions in the Hamiltonian limit. In addition, we find basin volume to remain essentially constant along a peculiar parameter path along which it is possible to recover the dissipation rate solely from metric properties of self-similar structures in phase-space.