Quasi-periodic Attractors Research Articles

Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds, Part II: Bifurcation and quasi-periodic response

In Part I of this paper, we have used spectral submanifold (SSM) theory to construct reduced-order models for harmonically excited mechanical systems with internal resonances. In that setting, extracting forced response curves formed by periodic orbits of the full system was reduced to locating the solution branches of equilibria of the corresponding reduced-order model. Here, we use bifurcations of the equilibria of the reduced-order model to predict bifurcations of the periodic response of the full system. Specifically, we identify Hopf bifurcations of equilibria and limit cycles in reduced models on SSMs to predict the existence of two-dimensional and three-dimensional quasi-periodic attractors and repellers in periodically forced mechanical systems of arbitrary dimension. We illustrate the accuracy and efficiency of these computations on finite-element models of beams and plates.

On occurrence of mixed-torus bursting oscillations induced by non-smoothness

The paper devotes to the slow–fast behaviors of a higher-dimensional non-smooth system with the coupling of two scales. Some novel bursting attractors and interesting phenomena are presented, especially the so-called mixed-torus bursting oscillations, in which the trajectory moves along different tori in turn, though the bursting attractor still behaves in quasi-periodic form. Based on a 4-D hyper-chaotic model with two scrolls, a modified non-smooth slow–fast version is established. Upon the smooth and non-smooth bifurcations analysis, the coexisted attractors with the variation of the slow-varying parameter are derived. With the increase in the exciting amplitude, bursting oscillations may change from periodic to quasi-periodic attractor, the mechanism of which is obtained by employing the overlap of the transformed phase portrait and coexisted attractors as well as the bifurcations. Sliding along the boundary on the trajectory can be observed on the bursting attractor, the mechanism of which can be accounted for by employing the non-smooth theory or by the in-turn influence of two pseudo-attractors in different regions. Super-cone defined by an unstable focus may lead to the period-adding or period-decreasing phenomena in the spiking oscillations, while fold limit cycle bifurcation may result in the dramatical change of the spiking amplitude. Different limit cycles may involve the full vector field, resulting in the so-called mixed-torus bursting oscillations, in which the quiescent states seem to be pivots of the oscillations.

On the dynamics of fractional q-deformation chaotic map

In this paper, the dynamical behaviors of fractional q-deformation chaotic map are analyzed. Firstly, the fractional q-deformation chaotic map is proposed by employing the Caputo delta difference operator. Secondly, the rich dynamical behaviors, such as numerically stable period (NSP) attractor, quasi-periodic attractor, strange nonchaotic attractor, and chaotic attractor, of the proposed map are discussed by utilizing bifurcation diagram, phase diagram, and 0–1 test. Thirdly, two controllers are designed to study the chaos control and synchronization of the fractional q-deformation chaotic map. Finally, numerical simulations are presented to demonstrate the findings.

Synchronization and chimeras in a network of four ring-coupled thermoacoustic oscillators

We take a complex systems approach to investigating experimentally the collective dynamics of a network of four self-excited thermoacoustic oscillators coupled in a ring. Using synchronization metrics, we find a wide variety of emergent multi-scale behaviour, such as (i) a transition from intermittent frequency locking on a $\mathbb {T}^{3}$ quasiperiodic attractor to a breathing chimera, (ii) a two-cluster state of anti-phase synchronization on a periodic limit cycle, and (iii) a weak anti-phase chimera. We then compute the cross-transitivity from recurrence networks to identify the dominant direction of the coupling between the heat-release-rate ($q^{\prime }_{\mathbb {X}}$) and pressure ($p^{\prime }_{\mathbb {X}}$) fluctuations in each individual oscillator, as well as that between the pressure ($p^{\prime }_{\mathbb {X}}$ and $p^{\prime }_{\mathbb {Y}}$) fluctuations in each pair of coupled oscillators. We find that networks of non-identical oscillators exhibit circumferentially biased $p^{\prime }_{\mathbb {X}}$–$p^{\prime }_{\mathbb {Y}}$ coupling, leading to mode localization, whereas networks of identical oscillators exhibit globally symmetric $p^{\prime }_{\mathbb {X}}$–$p^{\prime }_{\mathbb {Y}}$ coupling. In both types of networks, we find that the $p^{\prime }_{\mathbb {X}}$–$q^{\prime }_{\mathbb {X}}$ coupling can be symmetric or asymmetric, but that the asymmetry is always such that $q^{\prime }_{\mathbb {X}}$ exerts a greater influence on $p^{\prime }_{\mathbb {X}}$ than vice versa. Finally, we show through a cluster analysis that the $p^{\prime }_{\mathbb {X}}$–$p^{\prime }_{\mathbb {Y}}$ interactions play a more critical role than the $p^{\prime }_{\mathbb {X}}$–$q^{\prime }_{\mathbb {X}}$ interactions in defining the collective dynamics of the system. As well as providing new insight into the interplay between the $p^\prime_{\mathbb{X}}\text{--}p^\prime_{\mathbb{Y}}$ and $p^\prime_{\mathbb{X}}\text{--}q^\prime_{\mathbb{X}}$ coupling, this study shows that even a small network of four ring-coupled thermoacoustic oscillators can exhibit a wide variety of collective dynamics. In particular, we present the first evidence of chimera states in a minimal network of coupled thermoacoustic oscillators, paving the way for the application of oscillation quenching strategies based on chimera control.

Core motifs predict dynamic attractors in combinatorial threshold-linear networks.

Combinatorial threshold-linear networks (CTLNs) are a special class of inhibition-dominated TLNs defined from directed graphs. Like more general TLNs, they display a wide variety of nonlinear dynamics including multistability, limit cycles, quasiperiodic attractors, and chaos. In prior work, we have developed a detailed mathematical theory relating stable and unstable fixed points of CTLNs to graph-theoretic properties of the underlying network. Here we find that a special type of fixed points, corresponding to core motifs, are predictive of both static and dynamic attractors. Moreover, the attractors can be found by choosing initial conditions that are small perturbations of these fixed points. This motivates us to hypothesize that dynamic attractors of a network correspond to unstable fixed points supported on core motifs. We tested this hypothesis on a large family of directed graphs of size n = 5, and found remarkable agreement. Furthermore, we discovered that core motifs with similar embeddings give rise to nearly identical attractors. This allowed us to classify attractors based on structurally-defined graph families. Our results suggest that graphical properties of the connectivity can be used to predict a network's complex repertoire of nonlinear dynamics.

Cluster-based hierarchical network model of the fluidic pinball – cartographing transient and post-transient, multi-frequency, multi-attractor behaviour

We propose a self-supervised cluster-based hierarchical reduced-order modelling methodology to model and analyse the complex dynamics arising from a sequence of bifurcations for a two-dimensional incompressible flow of the fluidic pinball. The hierarchy is guided by a triple decomposition separating a slowly varying base flow, dominant shedding and secondary flow structures. All these flow components are kinematically resolved by a hierarchy of clusters. The transition dynamics between these clusters is described by a directed network, called the cluster-based hierarchical network model (HiCNM). Three consecutive Reynolds number regimes for different dynamics are considered: (i) periodic shedding at $Re=80$ , (ii) quasi-periodic shedding at $Re=105$ and (iii) chaotic shedding at $Re=130$ , involving three unstable fixed points, three limit cycles, two quasi-periodic attractors and a chaotic attractor. The HiCNM enables identification of the dynamics between multiple invariant sets in a self-supervised manner. Both the global trends and the local structures during the transition are well resolved by a moderate number of hierarchical clusters. The proposed HiCNM provides a visual representation of transient and post-transient, multi-frequency, multi-attractor behaviour and may automate the identification and analysis of complex dynamics with multiple scales and multiple invariant sets.

Routh–Hurwitz Stability and Quasiperiodic Attractors in a Fractional‐Order Model for Awareness Programs: Applications to COVID‐19 Pandemic

This work explores Routh–Hurwitz stability and complex dynamics in models for awareness programs to mitigate the spread of epidemics. Here, the investigated models are the integer‐order model for awareness programs and their corresponding fractional form. A non‐negative solution is shown to exist inside the globally attracting set (GAS) of the fractional model. It is also shown that the diseasefree steady state is locally asymptotically stable (LAS) given that R0 is less than one, where R0 is the basic reproduction number. However, as R0 > 1, an endemic steady state is created whose stability analysis is studied according to the extended fractional Routh–Hurwitz scheme, as the order lies in the interval (0,2]. Furthermore, the proposed awareness program models are numerically simulated based on the predictor‐corrector algorithm and some clinical data of the COVID‐19 pandemic in KSA. Besides, the model’s basic reproduction number in KSA is calculated using the selected data (R0 = 1.977828168). In conclusion, the findings indicate the effectiveness of fractional‐order calculus to simulate, predict, and control the spread of epidemiological diseases.

Transition to chaos in a reduced-order model of a shear layer

The present work studies the nonlinear dynamics of a shear layer, driven by a body force and confined between parallel walls, a simplified setting to study transitional and turbulent shear layers. It was introduced by Nogueira & Cavalieri (J. Fluid Mech., vol. 907, 2021, A32), and is here studied using a reduced-order model based on a Galerkin projection of the Navier–Stokes system. By considering a confined shear layer with free-slip boundary conditions on the walls, periodic boundary conditions in streamwise and spanwise directions may be used, simplifying the system and enabling the use of methods of dynamical systems theory. A basis of eight modes is used in the Galerkin projection, representing the mean flow, Kelvin–Helmholtz vortices, rolls, streaks and oblique waves, structures observed in the cited work, and also present in shear layers and jets. A dynamical system is obtained, and its transition to chaos is studied. Increasing Reynolds number $Re$ leads to pitchfork and Hopf bifurcations, and the latter leads to a limit cycle with amplitude modulation of vortices, as in the direct numerical simulations by Nogueira & Cavalieri. Further increase of $Re$ leads to the appearance of a chaotic saddle, followed by the emergence of quasi-periodic and chaotic attractors. The chaotic attractors suffer a merging crisis for higher $Re$ , leading to a chaotic dynamics with amplitude modulation and phase jumps of vortices. This is reminiscent of observations of coherent structures in turbulent jets, suggesting that the model represents a dynamics consistent with features of shear layers and jets.

Nontrivial amplitude death in coupled parity-time-symmetric Liénard oscillators.

We unravel the collective dynamics exhibited by two coupled nonlinearly damped Liénard oscillators exhibiting parity and time symmetry, which is a classical example of the position-dependent damped systems. The coupled system facilitates the onset of limit-cycle and aperiodic oscillations in addition to large-amplitude oscillations. In particular, a nontrivial amplitude death state emerges as a consequence of balanced linear loss and gain of the coupled PT-symmetric systems, where gain in the amplitude of oscillation in one oscillator is exactly balanced by the loss in the other. Further, quasiperiodic attractors exist in the parameter space of a neutrally stable trivial steady state. We deduce analytical critical curves enclosing the stable regions of a nontrivial fixed point, leading to the manifestation of nontrivial amplitude death state, and neutrally stable trivial steady state. The latter loses its stability leading to the emergence of the former. The analytical critical curves exactly match with the simulation boundaries. There is also a reemergence of dynamical states as a function of the coupling strength and multistability among the observed dynamical states. The basin of attraction provides an explanation for the observed probability of dynamical states.

KAM quasi-periodic tori for the dissipative spin–orbit problem

We provide evidence of the existence of KAM quasi-periodic attractors for a dissipative model in Celestial Mechanics. We compute the attractors extremely close to the breakdown threshold.We consider the spin–orbit problem describing the motion of a triaxial satellite around a central planet under the simplifying assumption that the center of mass of the satellite moves on a Keplerian orbit, the spin-axis is perpendicular to the orbit plane and coincides with the shortest physical axis. We also assume that the satellite is non-rigid; as a consequence, the problem is affected by a dissipative tidal torque that can be modeled as a time-dependent friction, which depends linearly upon the velocity.Our goal is to fix a frequency and compute the embedding of a smooth attractor with this frequency. This task requires to adjust a drift parameter.We have shown in Calleja et al. (2020) that it is numerically efficient to study Poincaré maps; the resulting spin–orbit map is conformally symplectic, namely it transforms the symplectic form into a multiple of itself. In Calleja et al. (2020), we have developed an extremely efficient (quadratically convergent, low storage requirements and low operation count per step) algorithm to construct quasi-periodic solutions and we have implemented it in extended precision. Furthermore, in Calleja et al. (2020) we have provided an “a-posteriori” KAM theorem that shows that if we have an embedding and a drift parameter that satisfy the invariance equation up to an error which is small enough with respect to some explicit condition numbers, then there is a true solution of the invariance equation. This a-posteriori result is based on a Nash–Moser hard implicit function theorem, since the Newton method incurs losses of derivatives.The goal of this paper is to provide numerical calculations of the condition numbers and verify that, when they are applied to the numerical solutions, they will lead to the existence of the torus for values of the parameters extremely close to the parameters of breakdown. Computing reliably close to the breakdown allows to discover several interesting phenomena, which we will report in Calleja et al. (2020).The numerical calculations of the condition numbers presented here are not completely rigorous, since we do not use interval arithmetic to estimate the round off error and we do not estimate rigorously the truncation error, but we implement the usual standards in numerical analysis (using extended precision, checking that the results are not affected by the level of precision, truncation, etc.). Hence, we do not claim a computer-assisted proof, but the verification is more convincing than a standard numerics. We hope that our work could stimulate a computer-assisted proof.

Finite-size coherent particle structures in high-Prandtl-number liquid bridges

Clustering of small rigid spherical particles in particle accumulation structures (PAS) of various shapes is found with highly resolved numerical simulations in a thermocapillary liquid bridge at a Prandtl number of 68. The intricate Kolmogorov--Arnold--Moser (KAM) structure of the fluid flow is unraveled for several Reynolds numbers for flows periodic in time and space. For these flows, a rich variety of periodic and quasiperiodic attractors for particles nearly density-matched to the fluid is found near closed streamlines and KAM tori approaching the free surface closely. A large parametric study finds the PAS dependence on particle size, particle-to-fluid density ratio, and Reynolds number.

Stochastic variability of regular and chaotic dynamics in 2D metapopulation model

A behavior of metapopulation consisting of two coupled subsystems modeled by the Ricker map is considered. We study how dynamics of the metapopulation changes under increase in the intensity of migration between subpopulations. For the deterministic model, a variety of equilibrium, periodic, quasiperiodic, and chaotic attractors is described. An impact of random disturbances on the behavior of metapopulation is studied both numerically and analytically with the help of confidence domains. A phenomenon of the noise-induced temporal stabilization of the unstable equilibrium is discovered. We point out the special role of transients and fractal riddled basins in the noise-induced transitions from order to chaos.

Destruction and resurgence of the quasiperiodic shearless attractor.

We consider a dissipative version of the standard nontwist map. It is known that nontwist systems may present a robust transport barrier, called shearless curve, that gives rise to an attractor that retains some of its properties when dissipation is introduced. This attractor is known as shearless attractor, and it may be quasiperiodic or chaotic depending on the control parameters. We describe a route for the destruction and resurgence of the quasiperiodic shearless attractor by analyzing the manifolds of the unstable periodic orbits (UPOs) which are fixed points of the map. We show that the shearless attractor is destroyed by a collision with the UPOs and it resurges after the reconnection of the unstable manifolds of different UPOs.

Dynamics of coupled modified Rössler oscillators: The role of nonisochronicity parameter.

The amplitude-dependent frequency of the oscillations, termed nonisochronicity, is one of the essential characteristics of nonlinear oscillators. In this paper, the dynamics of the Rössler oscillator in the presence of nonisochronicity is examined. In particular, we explore the appearance of a new fixed point and the emergence of a coexisting limit-cycle and quasiperiodic attractors. We also describe the sequence of bifurcations leading to synchronized, desynchronized attractors and oscillation death states in the coupled Rössler oscillators as a function of the strength of nonisochronicity and coupling parameters. Furthermore, we characterize the multistability of the coexisting attractors by plotting the basins of attraction. Our results open up the possibilities of understanding the emergence of coexisting attractors and into a qualitative change of the collective states in coupled nonlinear oscillators in the presence of nonisochronicity.

Chaotic transients, riddled basins, and stochastic transitions in coupled periodic logistic maps.

A system of two coupled map-based oscillators is studied. As units, we use identical logistic maps in two-periodic modes. In this system, increasing coupling strength significantly changes deterministic regimes of collective dynamics with coexisting periodic, quasiperiodic, and chaotic attractors. We study how random noise deforms these dynamical regimes in parameter zones of mono- and bistability, causes "order-chaos" transformations, and destroys regimes of in-phase and anti-phase synchronization. In the analytical study of these noise-induced phenomena, a stochastic sensitivity technique and a method of confidence domains for periodic and multi-band chaotic attractors are used. In this analysis, a key role of chaotic transients and geometry of "riddled" basins is revealed.

STRANGE NONCHAOTIC ATTRACTORS AND MULTISTABILITY IN A TWO-DEGREE-OF-FREEDOM QUASIPERIODICALLY FORCED VIBRO-IMPACT SYSTEM

In this work, we initially study the strange nonchaotic dynamics of a two-degree-of-freedom quasiperiodically forced vibro-impact system. It is shown that strange nonchaotic attractors (SNAs) occur between two chaotic regions, but not between the quasiperiodic region and the chaotic one. Subsequently, we mainly focus on the abundant multistability in the system, especially the coexistence of SNAs and quasiperiodic attractors. Besides, the coexistence of quasiperiodic attractors of different frequencies, and the coexistence of quasiperiodic attractors and chaotic attractors are also uncovered. The basins of attraction of these coexisting attractors are obtained. The quasiperiodic attractor can transform into a chaotic attractor directly through torus break-up without passing through an SNA. The nonchaotic property of SNAs is verified by its maximal Lyapunov exponent, and the strange property of SNAs is described by its phase sensitivity, power spectrum, fractal structure, and rational approximations.

Multistability and chaos in a noise-induced blood flow

The present study introduces an analysis framework to understand the dynamics of a two-dimensional non-autonomous coronary artery model. The analysis shows that the coronary artery system exhibits chaotic attractors, quasi-periodic, and periodic orbits. For a specific set of parameters, an asymptotic periodic orbit appears in the short duration time and subsequently a wavy periodic orbit in the long-term. Besides, by choosing the initial conditions near a boundary of the basins of attraction, we observe a complex transition from transient chaos to the quasi-periodic attractor. More interestingly, the striking dynamical behaviors of coexisting two 2-cycles, symmetric chaotic attractors, and quasi-periodic beside chaotic attractors observe by selecting appropriate sets of initial conditions. The presence of coexisting attractors reflects the high sensitivity of the system. This fact likewise is confirmed by the Sample entropy algorithm, which depicts the variety of complexity values as the initial conditions varying. To investigate the effect of random noise on the system, we discuss the stochastic fluctuation in the birhythmicity region and the noise-induced transition using the 0–1 test.

Dynamics of a Heterogeneous Constraint Profit Maximization Duopoly Model Based on an Isoelastic Demand

A Cournot duopoly game is a two-firm market where the aim is to maximize profits. It is rational for every company to maximize its profits with minimal sales constraints. As a consequence, a model of constrained profit maximization (CPM) occurs when a business needs to be increased with profit minimal sales constraints. The CPM model, in which companies maximize profits under the minimum sales constraints, is an alternative to the profit maximization model. The current study constructs a duopoly game based on an isoelastic demand and homogeneous goods with heterogeneous strategies. In the event of sales constraint and no sales constraint, the local stability conditions of the Cournot equilibrium are derived. The initial results show that the duopoly model would be easier to stabilize if firms were to impose certain minimum sales constraints. Two routes to chaos are analyzed by numerical simulation using 2D bifurcation diagram, one of which is period doubling bifurcation and the other is Neimark–Sacker bifurcation. Four forms of coexistence of attractors are demonstrated by the basin of attraction, which is the coexistence of periodic attractors and chaotic attractors, the coexistence of periodic attractors and quasiperiodic attractors, and the coexistence of several chaotic attractors. Our findings show that the effect of game parameters on stability depends on the rules of expectations and restriction of sales by firms.

DYNAMICAL ANALYSIS OF THE MEMINDUCTOR-BASED CHAOTIC SYSTEM WITH HIDDEN ATTRACTOR

The meminductor is a new type of memory circuit element which is defined based on the memristor. To explore the application of the meminductor in the nonlinear circuits, a mathematical model of meminductor is proposed and applied to nonlinear circuits. In this work, a simple meminductor-based chaotic system is designed. The equilibrium point of the system is controlled by the externally excited sinusoidal signal in the circuit. No matter what the value of the externally excited signal is, the chaotic attractor generated by the proposed system is hidden. The dynamic characteristics of the system are analyzed by theoretical analysis and numerical simulation. The results show that the dynamic behaviors of the system are affected by the circuit parameters and the circuit running time. The proposed system shows some novel nonlinear phenomena, such as transient chaos and state transitions. In addition, the existence of coexisting attractors, such as chaotic, periodic and quasi-periodic attractors, is analyzed in different initial states.

Hidden Strange Nonchaotic Attractors

In this paper, it is found numerically that the previously found hidden chaotic attractors of the Rabinovich–Fabrikant system actually present the characteristics of strange nonchaotic attractors. For a range of the bifurcation parameter, the hidden attractor is manifestly fractal with aperiodic dynamics, and even the finite-time largest Lyapunov exponent, a measure of trajectory separation with nearby initial conditions, is negative. To verify these characteristics numerically, the finite-time Lyapunov exponents, ‘0-1’ test, power spectra density, and recurrence plot are used. Beside the considered hidden strange nonchaotic attractor, a self-excited chaotic attractor and a quasiperiodic attractor of the Rabinovich–Fabrikant system are comparatively analyzed.