Strange Attractors Research Articles

Singular strange attractors beyond the boundary of hyperbolic flows

We prove that the suspension of an expanding attractor in the two-disk can be deformed into a robust singular strange attractor. The deformation is done through an arc of three-dimensional flows with only one bifurcation. The attractors so obtained are not Lorenz-like.

A Fractal Approach to Nonlinear Topographical Features of Healthy and Keratoconus Corneas Pre- and Post-Operation of Intracorneal Implants

Fractal dimension (FD) together with advances in imaging technologies has provided an increasing application of digital images to interpret biological phenomena. In ophthalmology, topography-based images are increasingly used in common practices of clinical settings. They provide detailed information about corneal surfaces. Few-micron alterations of the corneal geometry to the elevation and curvature cause a highly multifocal surface, change the corneal optical power up to several diopters, and therefore adversely affect the individual’s vision. Keratoconus (KCN) is a corneal disease characterized by a local alteration of the corneal anatomical and mechanical features. The formation of cone-shaped regions accompanied by thinning and weakening of the cornea are the major manifestations of KCN. The implantation of tiny arc-like polymeric sections, known as intracorneal implants, is considered to be effective in restoring the corneal curvature. This study investigated the FD nature of healthy corneas (n = 7) and compared it to the corresponding values before and after intracorneal implant surgery in KCN patients (n = 7). The generalized Hurst exponent, Higuchi, and Katz FDs were computed for topography-based parameters of corneal surfaces: front elevation (ELE-front), back elevation (ELE-back), and corneal curvature (CURV). The Katz FD showed better discriminating ability for the diseased group. It could reveal a significant difference between the healthy corneas and both pre- and post-implantation topographies (p < 0.001). Moreover, the Katz dimension varied between the topographic features of KCN patients before and after the treatment (p < 0.036). We propose to describe the curvature feature of corneal topography as a “strange attractor” with a self-similar (i.e., fractal) structure according to the Katz algorithm.

Mathematical modeling of the evolution of a tribosystem in the process of formation of equilibrium roughness of a frictional contact

As the science of friction developed, tribologists showed that reliable operation of friction units is determined by the wear resistance of the surface layer, its ability to adapt to changing operating conditions. In this regard, there is a need not only to create structures on the surfaces of friction units that have the properties of self-regulation and self-organization, but also to monitor changes in the characteristics of the surface layer. This article presents an innovative methodology for monitoring the life cycle of surface layers coated by vacuum ion-plasma treatment. In this paper, we propose a dynamic model based on the hypothesis that roughness is both a factor that excites mechanical vibrations and a consequence of these vibrations, whereas roughness changes as a result of wear under the influence of local dynamic forces. It is shown that a friction pair is a nonlinear system with complex feedbacks. Its stationary motion can be considered as a strange attractor. The randomness of motion in the vicinity of the limit cycle is due to the presence of a link of constant delay with rigid positive feedback in the model. As a result, there is constant reconfiguration of the amplitude and phase of the oscillations.

Verification and accuracy check of simulations with PoPe and iPoPe

The theoretical background of the PoPe and iPoPe verification scheme is presented. Verification is performed using the simulation output of production runs. The computing overhead is estimated to be at most 10%. PoPe or iPoPe calculations can be done offline provided the necessary data is stored, for example additional time slices, or online where iPoPe is more effective. The computing overhead is mostly that of storing the necessary data. The numerical error is determined and split into a part proportional to the operators, which are combined to form the equations to be solved, thus modifying their control parameters, completed by a residual error orthogonal to these operators. The accuracy of the numerical solution is determined by this modification of the control parameters. The PoPe and iPoPe methods are illustrated in this paper with simulations of a simple mechanical system with chaotic trajectories evolving into a strange attractor with sensitivity to initial conditions. We show that the accuracy depends on the particular simulation both because the properties of the numerical solution depend on the values of the control parameter, and because the target accuracy will depend on the problem that is addressed. One shows that for a case close to bifurcations between different states, the accuracy is determined by the level of detail of the bifurcation phenomena one aims at describing. A unique verification index, the PoPe index, is proposed to characterise the accuracy, and consequently the verification, of each production run. The PoPe output allows one to step beyond verification and analyse for example the numerical scheme efficiency. For the chosen example at fixed PoPe index, therefore at fixed numerical error, one finds that the higher order integration scheme, comparing order 4 to order 2 Runge-Kutta time stepping, reduces the computation cost by a factor 4.

Generic generation of noise-driven chaos in stochastic time delay systems: Bridging the gap with high-end simulations

Nonlinear time delay systems produce inherently delay-induced periodic oscillations, which are, however, too idealistic compared to observations. We exhibit a unified stochastic framework to systematically rectify such oscillations into oscillatory patterns with enriched temporal variabilities through generic, nonlinear responses to stochastic perturbations. Two paradigms of noise-driven chaos in high dimension are identified, fundamentally different from chaos triggered by parameter-space noise. Noteworthy is a low-dimensional stretch-and-fold mechanism, leading to stochastic strange attractors exhibiting horseshoe-like structures mirroring turbulent transport of passive tracers. The other is high-dimensional , with noise acting along the critical eigendirection and transmitted to "deeper" stable modes through nonlinearity, leading to stochastic attractors exhibiting swarm-like behaviors with power-law and scale break properties. The theory is applied to cloud delay models to parameterize missing physics such as intermittent rain and Lagrangian turbulent effects. The stochastically rectified model reproduces with fidelity complex temporal variabilities of open-cell oscillations exhibited by high-end cloud simulations.

Dynamical Behaviors of an SIR Epidemic Model with Discrete Time

Analytically and numerically, the study examines the stability and local bifurcations of a discrete-time SIR epidemic model. For this model, a number of bifurcations are studied, including the transcritical, flip bifurcations, Neimark–Sacker bifurcations, and strong resonances. These bifurcations are checked, and their non-degeneracy conditions are determined by using the normal form technique (computing of critical normal form coefficients). We use the MATLAB toolbox MatcontM, which is based on the numerical continuation method, to confirm the obtained analytical results and specify more complex behaviors of the model. Numerical simulation is employed to present a closed invariant curve emerging from a Neimark–Sacker point and its breaking down to several closed invariant curves and eventually giving rise to a chaotic strange attractor by increasing the bifurcation parameter.

Phase-locking flows between orthogonally stretching parallel plates

In this paper, we explore the stability and dynamical relevance of a wide variety of steady, time-periodic, quasiperiodic, and chaotic flows arising between orthogonally stretching parallel plates. We first explore the stability of all the steady flow solution families formerly identified by Ayats et al. [“Flows between orthogonally stretching parallel plates,” Phys. Fluids 33, 024103 (2021)], concluding that only the one that originates from the Stokesian approximation is actually stable. When both plates are shrinking at identical or nearly the same deceleration rates, this Stokesian flow exhibits a Hopf bifurcation that leads to stable time-periodic regimes. The resulting time-periodic orbits or flows are tracked for different Reynolds numbers and stretching rates while monitoring their Floquet exponents to identify secondary instabilities. It is found that these time-periodic flows also exhibit Neimark–Sacker bifurcations, generating stable quasiperiodic flows (tori) that may sometimes give rise to chaotic dynamics through a Ruelle–Takens–Newhouse scenario. However, chaotic dynamics is unusually observed, as the quasiperiodic flows generally become phase-locked through a resonance mechanism before a strange attractor may arise, thus restoring the time-periodicity of the flow. In this work, we have identified and tracked four different resonance regions, also known as Arnold tongues or horns. In particular, the 1 : 4 strong resonance region is explored in great detail, where the identified scenarios are in very good agreement with normal form theory.

Ensemble data assimilation using optimal control in the Wasserstein metric

An ensemble data assimilation method is proposed that is based on optimal control minimizing the cost of mismatch in the Wasserstein metric on the observation space. The new method achieved the optimal state without calculating the posterior distribution of the particle state and the particle states are evolved deterministically, which is easy to be implemented. The method is appropriate for systems in which multiple, noisy, partial observations are available (e.g. citizen weather stations or smart phones). The method is demonstrated for: (i) deterministic dynamics with uncertain initial conditions, (ii) multiple noisy observations of a randomly forced ordinary differential equation (ODE), (iii) observations from multiple sample paths from a stochastic differential equation (SDE). A bi-modal measure and a measure supported on a strange attractor are tested. The numerical results show that our method performs a relatively small Wasserstein distance which measures the approximation performance. But numerical implementation is a bit expensive due to the complexity of Wasserstein distance computation, especially with large set of particles.

Communication management in a millennium of chaos and change

Changes occur in organisations because of changes in the environment.Conflicts arise between the organisation and internal and externalstakeholders of the organisation because of these uncontrolledinfluences. Small points of disorder may easily escalate into disorderedcrises and chaos (bifurcations and the butterfly effect). Communicationmanagement - and specifically constructive conflict management - couldlead to 'positive chaos' and a culture (strange attractor) of constantchange. Positive chaos implies an unstable, seemingly disorderedsituation that could bring about productive, creative, and improvedresults. Communication management could be used by organisations toreorganise themselves out of disorder and chaos by facilitating conflictsand diversity. Strategic planning will become even more important, butwill have to change from the basic premise of control and prediction toscenario planning, and the emphasis will be on relationship building.

Characteristics of the Invariant Measure of the Strange Attractor of the Bacteria Mathematical Model

The bacteria metabolic process of open nonlinear dissipative system far from equilibrium point is modeled using classical methods of synergetics. The invariant measure and its convergence in the phase space of the system was obtained in strange attractor mode. The distribution of point density of trajectory intersection of phase space cells with maximum invariant measure and convergence in time of its average value was obtained. The result concluded is that the value of an invariant measure can be a characteristic of the transitional process of adaptation of cell metabolic process to change outside environment.

Stochastic chaos in chemical Lorenz system: Interplay of intrinsic noise and nonlinearity

In this paper, we address the question of whether the phenomenon of chaos can occur in a purely stochastic system or not. We show that chaos can arise in a purely stochastic chemical Lorenz system. A robust regime of stochastic chaos develops like the onset of chaos in a deterministic system. Stochastic trajectories, which are initially very close, show sensitivity to initial states where they diverge exponentially as time progress. The interplay of nonlinearity and intrinsic noise in the chemical Lorenz system is studied to understand the effect of intrinsic noise. We observe that intrinsic noise can destabilize the fixed points and limit cycle attractors where stochastic trajectories make excursions along the unstable manifold, giving rise to the noisy chaotic attractors. This study uses quantitative measures like power spectrums and invariant measures to characterize the strange noisy attractors. Our study establishes that the interplay of intrinsic noise and nonlinearity gives rise to chaos in the stochastic Lorenz system.

Nonlinear dynamics of a heterogeneous quantum Commons’ tragedy

This paper first builds a nonlinear dynamic system of the Commons’ tragedy with quantum strategies in discrete time. There are two heterogeneous expectations in the game, namely, one of the two players is boundedly rational and the other is of adaptive expectation. Secondly, we obtain a sufficient and necessary condition for quantum Nash equilibria to be stable equilibria. Finally, we investigate the impacts of quantum entanglement on the stable region of systems and analyze the behaviors of complex dynamical systems. The results show that quantum Nash equilibria are stable points of nonlinear dynamics for quantum Commons’ tragedy if both quantum entanglement and quantum strategy adjustment speed satisfy certain conditions. In addition, numerical simulations discuss the features of nonlinear dynamics, such as bifurcation, the largest Lyapunov exponent, fractal dimension, strange attractor, and sensitivity to initial conditions. The numerical results show that nonlinear dynamics appear chaotic if the adjustment speed of quantum strategy or the quantum entanglement degree is larger.

Birth of strange nonchaotic attractors in a piecewise linear oscillator.

Nonsmooth systems are widely encountered in engineering fields. They have abundant dynamical phenomena, including some results on the complex dynamics in such systems under quasiperiodically forced excitations. In this work, we consider a quasiperiodically forced piecewise linear oscillator and show that strange nonchaotic attractors (SNAs) do exist in such nonsmooth systems. The generation and evolution mechanisms of SNAs are discussed. The torus-doubling, fractal, bubbling, and intermittency routes to SNAs are identified. The strange properties of SNAs are characterized with the aid of the phase sensitivity function, singular continuous spectrum, rational frequency approximation, and the path of the partial Fourier sum of state variables in a complex plane. The nonchaotic properties of SNAs are verified by the methods of maximum Lyapunov exponent and power spectrum.

A simple three-dimensional quadratic flow with an attracting torus

Attracting torus is a rare phenomenon in the dynamics of low-dimensional autonomous systems. Adding an anti-damping term to the well-known Nosé-Hoover oscillator, this paper introduces a new system exhibiting attracting torus in a wide range of parameter values. This system has a variety of dynamical solutions like limit cycles, strange attractors, attracting tori, invariant tori, and chaotic sea. It is also demonstrated that the system is multistable in some regions of parameter space wherein different types of attractors coexist. However, the attracting torus is the leading bounded solution in a considerable area of parameter space. Moreover, the coexistence of four limit cycles is found in the time-reversed system. The study of the system's basin of attraction shows that the system owns a solid basin of attraction with rounded boundaries for the attracting torus, which is an exciting property.

On the origin of chaotic attractors with two zero Lyapunov exponents in a system of five biharmonically coupled phase oscillators.

We study chaotic dynamics in a system of four differential equations describing the interaction of five identical phase oscillators coupled via biharmonic function. We show that this system exhibits strange spiral attractors (Shilnikov attractors) with two zero (indistinguishable from zero in numerics) Lyapunov exponents in a wide region of the parameter space. We explain this phenomenon by means of bifurcation analysis of a three-dimensional Poincaré map for the system under consideration. We show that chaotic dynamics develop here near a codimension three bifurcation, when a periodic orbit (fixed point of the Poincaré map) has the triplet of multipliers ( 1 , 1 , 1 ). As it is known, the flow normal form for such bifurcation is the well-known three-dimensional Arneodó-Coullet-Spiegel-Tresser (ACST) system, which exhibits spiral attractors. According to this, we conclude that the additional zero Lyapunov exponent for orbits in the observed attractors appears due to the fact that the corresponding three-dimensional Poincaré map is very close to the time-shift map of the ACST-system.

The Mesoproterozoic Nampula Subdomain in southern Malawi: Completing the story from Mozambique

New field mapping, lithostratigraphic, whole-rock geochemical, U–Pb zircon and radiogenic isotope data on the Nampula Subdomain of southernmost Malawi show good correlation with those published from adjacent NE and NW Mozambique. The oldest rocks, dated between ∼1180 and 1070 Ma form a complexly interlayered sequence of supracrustal layered paragneisses and migmatites (Nampula Group), intimately interlayered with intermediate-basic (Mocuba Suite) and felsic (Mamala Suite) gneisses, likely meta-volcanic and meta-plutonic in origin. The Mocuba Suite has typical calc-alkaline, I-Type granitoid, TTG compositions, with generally positive ϵNd values, NdTdm ages around 1.2 Ga and low initial 87Sr/86Sr ratios. These characteristics indicate generation in a juvenile volcanic arc setting at this time. Minor exceptions include a few samples which show limited input of Palaeoproterozoic crust. The Nampula arc is viewed as one of a number that developed in a closing oceanic basin which lay between the Archaean-Palaeoproterozoic Congo-Bangweulu-Ubendian-Tanzania cratonic blocks in the north and the Zimbabwe Craton to the SW, and which can be traced “southwards” in Gondwana to continue in the NE part of the ∼1 Ga Maud belt in East Antarctica. The Nampula arc-complex was intruded at ∼1040 by syn-to late tectonic sheets of megacrystic granite/augen gneisses of the Culicui Suite and, at ∼1000 Ma, by a number of bodies of late-to post-tectonic, massif-type anorthosite (Tengani Suite), which are unique to the Malawian segment of the Nampula Subdomain. The anorthosites are considered to have intruded at a post-tectonic stage under a late Mesoproterozoic extensional regime. The rocks underwent deformation and high grade metamorphism in both the Mesoproterozoic (Irumide) and Neoproterozoic-Cambrian (Pan-African) orogenies. P-T data suggest the latter attained peak granulite facies conditions of ∼800 °C at a maximum of 13–18 kbar, comparable with Neoproterozoic granulites in adjacent areas. The data presented here do not rule out either Irumide (∼1 Ga) juxtaposition of the Congo and Kalahari cratons (in Rodinia), or Pan-African (∼0.5 Ga) collision (Gondwana). Instead of this “either-or” situation, a recent hypothesis suggests accretion/collision occurred during both orogenies, separated by a period of drift apart, by the “strange attractor” phenomenon. Our data do not rule exclude that contention; rather, they tend to support it.

Stability, Bifurcation, and Chaos Control of Two-Sided Market Competition

Benefitting from the popular uses of internet technologies, two-sided market has been playing an increasing prominent role in modern times. Users and developers can interact with each other through two-sided platforms. The two-sided market structure has been investigated profoundly. Through building a dynamics two-sided market model with bounded rational, stability conditions of the two-sided market competition system are presented. With the help of bifurcation diagram, Lyapunov exponent, and strange attractor, the stability of the two-sided market competition model is simulated. At last, we use the time-delayed feedback control (TDFC) method to control the chaos. Our main results are as follows: (1) when the adjustment speed of two-sided increases, the system becomes bifurcation, and chaos state happens finally. When the system is stable, the consumer fee is positive while developer fee is negative. (2) When the user externality increases, the stable area of the system increases, and the difference in user externality leads the whole system more stable. When the system is stable, the developer fee decreases. (3) The stable area becomes larger when developer externality increases; when the system is stable, the user fee becomes lower and developer fee becomes higher when developer externality increases. (4) The TDFC method is presented for controlling the chaos; we find that the system becomes more stable under the TDFC method.

Manifestation of strange nonchaotic attractors in extended systems: a study through out-of-time-ordered correlators

We study the spatial spread of out-of-time-ordered correlators (OTOCs) in coupled map lattices (CMLs) of quasiperiodically forced nonlinear maps. We use instantaneous speed (IS) and finite-time Lyapunov exponents (FTLEs) to investigate the role of strange non-chaotic attractors (SNAs) on the spatial spread of the OTOC. We find that these CMLs exhibit a characteristic on and off type of spread of the OTOC for SNA. Further, we provide a broad spectrum of the various dynamical regimes in a two-parameter phase diagram using IS and FTLEs. We substantiate our results by confirming the presence of SNA using established tools and measures, namely the distribution of finite-time Lyapunov exponents, phase sensitivity, spectrum of partial Fourier sums, and $0-1$ test.

Resonant three-wave interactions with linear source/sink terms: Exploring the parameter space

It is shown that the resonance three wave interactions in the presence of linear source/sink terms exhibit a wide variety of time-evolution dynamics, such as exponential growth, limit cycles, and strange attractors. The particular features of the dynamics are determined primarily by the values of linear growth rates, forming a 2D parameter space. An exploration of this parameter space is the main focus of this present work.

Dynamics of a class of Chua’s oscillator with a smooth periodic nonlinearity: Occurrence of infinitely many attractors

In this article, a class of Chua’s system with smooth periodic nonlinear term is considered. This kind of systems exhibits strange dynamical behavior. Except for the existence of infinitely many equilibrium points, period-doubling bifurcation, and double-scroll attractors, this type of systems has strange dynamics different from the classical Chua’s system: (i) the coexistence of (infinitely) many non-chaotic strange attractors; (ii) the coexistence of (infinitely) many spiral chaotic attractors; (iii) the coexistence of multi-scroll attractors; (iv) “growing”-scroll attractors, where the number of scrolls is an increasing function with respect to the time variable.