Aperiodic Trajectories Research Articles

Nonlinear model reduction to temporally aperiodic spectral submanifolds.

We extend the theory of spectral submanifolds (SSMs) to general non-autonomous dynamical systems that are either weakly forced or slowly varying. Examples of such systems arise in structural dynamics, fluid-structure interactions, and control problems. The time-dependent SSMs we construct under these assumptions are normally hyperbolic and hence will persist for larger forcing and faster time dependence that are beyond the reach of our precise existence theory. For this reason, we also derive formal asymptotic expansions that, under explicitly verifiable nonresonance conditions, approximate SSMs and their aperiodic anchor trajectories accurately for stronger, faster, or even temporally discontinuous forcing. Reducing the dynamical system to these persisting SSMs provides a mathematically justified model- reduction technique for non-autonomous physical systems whose time dependence is moderate either in magnitude or speed. We illustrate the existence, persistence, and computation of temporally aperiodic SSMs in mechanical examples under chaotic forcing.

A memory-based approach to model glorious uncertainties of love.

We propose a minimal yet intriguing model for a relationship between two individuals. The feeling of an individual is modeled by a complex variable and, hence, has two degrees of freedom [Jafari et al., Nonlinear Dyn. 83, 615-622 (2016)]. The effect of memory of the other individual's behavior in the past has now been incorporated via a conjugate coupling between each other's feelings. A region of parameter space exhibits multi-stable solutions wherein trajectories with different initial conditions end up in different aperiodic trajectories. This aligns with the natural observation that most relationships are aperiodic and unique not only to themselves but, more importantly, to the initial conditions too. Thus, the inclusion of memory makes the task of predicting the trajectory of a relationship hopelessly impossible.

A two-stroke growth cycle model for a small open economy

The Hicksian multiplier–accelerator model with a floor and/or a ceiling stands as one of the most successful representations of the economy capable of generating endogenous business cycles. In the present paper, we take some of the main insights of this literature as starting points to develop a continuous-time growth-cycle model for a small open economy. Our main innovation lies in assuming that capital accumulation is constrained only from above by the growth rate compatible with equilibrium in the balance-of-payments, i.e. the dynamic Harrod trade multiplier. It is shown that the obtained piecewise second order differential equation is a so-called two-stroke oscillator. Conditions under which the model produces periodic or aperiodic trajectories are studied analytically and through numerical simulations. An initial assessment of the statistical properties of the time-series indicates that they exhibit high kurtosis and negative skewness.

Dynamics of a two-dimensional map on nested circles and rings

• We consider a two-dimensional map whose bounded dynamics can be restricted to an invariant circle, cyclic invariant circles, invariant annual regions or disks. • We prove that on such invariant sets the trajectories are always either periodic of the same period, or quasiperiodic and dense. • We show that the invariant sets may be transversely attracting or repelling, and undergo the typical cascade of period doubling bifurcations. • We show that homoclinic bifurcations lead to chaotic rings, annual regions filled with dense repelling cyclical circles and aperiodic trajectories. We consider a discrete dynamical system, a two-dimensional real map which represents a one-dimensional complex map. Depending on the parameters, its bounded dynamics can be restricted to an invariant circle, cyclic invariant circles, invariant annular regions or disks. We show that on such invariant sets the trajectories are always either periodic of the same period, or quasiperiodic and dense. Moreover, the invariant sets may be transversely attracting or repelling, and undergo the typical cascade of period doubling bifurcations. Homoclinic bifurcations can also occur, leading to chaotic rings, annular regions filled with dense repelling cyclical circles and aperiodic trajectories.

Large-amplitude membrane flutter in inviscid flow

We study the large-amplitude flutter of membranes (of zero bending rigidity) with vortex sheet wakes in two-dimensional inviscid fluid flows. We apply small initial deflections and track their exponential decay or growth and subsequent large-amplitude dynamics in the space of three dimensionless parameters: membrane pretension, mass density and stretching modulus. With both ends fixed, all the membranes converge to steady deflected shapes with single humps that are nearly fore-aft symmetric, except when the deformations are unrealistically large. With leading edges fixed and trailing edges free to move in the transverse direction, the membranes flutter periodically at intermediate values of mass density. As mass density increases, the motions are increasingly aperiodic, and the amplitudes increase and spatial and temporal frequencies decrease. As mass density decreases from the periodic regime, the amplitudes decrease and spatial and temporal frequencies increase until the motions become difficult to resolve numerically. With both edges free to move in the transverse direction, the membranes flutter similarly to the fixed–free case, but also translate vertically with steady, periodic or aperiodic trajectories, and with non-zero slopes that lead to small angles of attack with respect to the oncoming flow.

Внешние биллиарды вне правильного десятиугольника: периодичность почти всех орбит и существование апериодической орбиты

Внешние биллиарды были введены Б. Нойманном в 50-х годах ХХ века и стали популярны в 70-х благодаря Ю. Мозеру, который рассматривал внешний, или двойственный, биллиард как игрушечную модель небесной механики. Задача об устойчивости Солнечной системы обладает тем свойством, что "легко выписать n уравнений движения частиц, но сложно понять это движение интуитивно"; в связи с этим, Мозер предложил рассмотреть ранее поставленную Б. Нойманном задачу внешнего биллиарда, обладающую тем же свойством.Одним из классических примеров динамической систем является внешний биллиард вне правильного ????-угольника; в частности, с ним связаны проблемы существования апериодической траектории, а также полноты периодических точек. Эти проблемы решены лишь для ограниченного количества частных случаев.При ???? = 3, 4, 6 стол является решеточным, и, как следствие, апериодических точек нет, а периодические точки образуют множество полной меры. В 1993 году, С. Табачникову удалось найти апериодическую точку в случае правильного пятиугольника; сделано это было с помощью ренормализационной схемы - метода, имеющего фундаментальное значение при исследовании самоподобных динамических систем.По мнению Р. Шварца, следующими по сложности являются случаи n = 10,8,12; в этих случаях, а также в случае ???? = 5 для внешнего биллиарда удается построить ренормализационную схему, которая, как пишет Шварц, “позволяет дать (как минимум, в принципе) полное описание того, что происходит”.Позже, автору удалось обнаружить самоподобные структуры и построить ренормализационную схему для случаев правильных восьми- и двенадцатиугольника.Данная же статья посвящена внешнему биллиарду вне правильного десятиугольника. Доказано существование апериодической орбиты для внешнего биллиарда вне правильного десятиугольника, а также, что почти все траектории такого внешнего биллиарда являются периодическими; явно выписаны все возможные периоды. В основе работы лежит классическая технология поиска и исследования ренормализационной схемы. Возникающие в случае ???? = 10 периодические структуры похожи на периодические структуры в случае ???? = 5, но все же имеют свои особенности.

Designing multi-dimensional logistic map with fixed-point finite precision

In cryptographic algorithms, random sequences of longer period and higher nonlinearity are always desirable in order to increase resistance against cryptanalysis. The use of chaotic maps is an attractive choice as they exhibit properties that are suitable for cryptography. In continuous phase space of the logistic map, proper control parameters and initial state result into aperiodic trajectories. However, when the phase space of the logistic map is quantized, the trajectories terminate in finite and stable periodic orbits due to quantization error. The dynamic degradation of the logistic map can be mitigated using nonlinear feedback and cascading multiple chaotic maps. We propose a logistic map-based, finite precision multi-dimensional logistic map, that incorporates nonlinear feedback and modulus operations to perturb the chaotic trajectories. We present complexity, average cycle length and randomness analysis to evaluate the proposed method. The simulation results and analysis reveal that the proposed MDLM approach achieves longer period and higher randomness.

Pace and patterns of magnetic swimmers in a billiard pool.

We experimentally investigate magnetic surface swimmers on water. These objects self-assemble from ferromagnetic microparticles and a nonmagnetic disk. They are floating on the liquid surface due to interface tension and move under the influence of a harmonically oscillating homogeneous magnetic field oriented vertically, which is distinguished by its amplitude and frequency. The speed of the surface swimmers strongly depends on these parameters. The functional dependencies between speed and amplitude and between speed and frequency are investigated by independently varying both control parameters. In the first case, the data obtained are in good agreement with the predicted scaling while there are some deviations in the latter case. Moreover, due to the interplay between the surface bound swimmers and the ascending liquid meniscus at the edge of the experimental vessel, different dynamics can be realized. We observe periodic and quasiperiodic trajectories in a circular vessel and aperiodic trajectories in a vessel shaped like a Bunimovich stadium.

A Novel Image Encryption Scheme Based on Clifford Attractor and Noisy Logistic Map for Secure Transferring Images in Navy

Cryptography is a science to maintain the security of the message by changing data or information into a different form, so the message cannot be recognized. Today, many algorithms have been proposed for image encryption, but the chaotic encryption methods have a good combination of speed and high security. In recent years, the chaos based cryptographic algorithms have suggested some new and efficient ways to develop secure image encryption techniques. The chaos-based encryption schemes are composed of two steps: chaotic confusion and pixel diffusion. In the chaotic confusion stage, a combination of the chaotic maps is used to realize the confusion of all pixels.In this paper, we first give a brief introduction into chaotic image encryption and then we investigate some important properties and behaviour of the logistic map. The logistic map, aperiodic trajectory, or random-like fluctuation, could not be obtained with some choice of initial condition. Therefore, a noisy logistic map with an additive system noise is introduced. The proposed scheme is based on the extended map of the Clifford strange attractor, where each dimension has a specific role in the encryption process. Two dimensions are used for pixel permutation and the third dimension is used for pixel diffusion. In order to optimize the Clifford encryption system we increase the space key by using the noisy logistic map and a novel encryption scheme based on the Clifford attractor and the noisy logistic map for secure transfer images is proposed. This algorithm consists of two parts: the noisy logistic map shuffle of the pixel position and the pixel value. We use times for shuffling the pixel position and value then we generate the new pixel position and value by the Clifford system. To illustrate the efficiency of the proposed scheme, various types of security analysis are tested. It can be concluded that the proposed image encryption system is a suitable choice for practical applications.

Emergence of chaos in interacting communities

We introduce a simple dynamical model of two interacting communities whose elements are subject to stochastic discrete-time updates governed by only bilinear interactions. When the intra- and inter-couplings are cooperative, the two communities reach asymptotically an equilibrium state. However, when the intra- or inter-couplings are anti-cooperative, the system may remain in perpetual oscillations and, when the coupling values belong to certain intervals, two possible scenarios arise, characterized either by erratic aperiodic trajectories and high sensitiveness to small changes of the couplings, or by chaotic trajectories and bifurcation cascades. Quite interestingly, we find out that even a moderate consensus in one single community can remove the chaos. Connections of the model with interacting stock markets are discussed.

Complex dynamics occur in a single-locus, multiallelic model of general frequency-dependent selection

We examine the characteristics of non-equilibrium dynamics produced by a simple well-known model of frequency-dependent selection at a single diploid locus. An examination of the parameter space of this “pairwise-interaction model” (PIM) revealed non-equilibrium dynamics for polymorphisms of 3, 4 and 5 alleles; both allele-frequency cycling and aperiodic trajectories were detected. We measured the number, cycle length and domains of attraction of the various attractors produced by the model. The domains of attraction tended to be smaller, and the cycles longer, for systems with larger number of alleles. Fitnesses that parametrized negative frequency-dependent selection were more likely to allow cycling, and these cycles also had larger domains of attraction. Aperiodic trajectories were detected only in cases with 4 or 5 alleles. The genetic cycles produced by the model do not have periods as short as those predicted in ecological models with cycling (such as predator–prey population cycles, etc.). Consequently, in a real-world system, PIM allele-frequency cycling is likely to be indistinguishable from stable equilibria when observed over short time scales.

Topological Entropy of Braids on the Torus

A fast method is presented for computing the topological entropy of braids on the torus. This work is motivated by the need to analyze large braids when studying two-dimensional flows via the braiding of a large number of particle trajectories. Our approach is a generalization of Moussafir's technique for braids on the sphere. Previous methods for computing topological entropies include the Bestvina--Handel train-track algorithm and matrix representations of the braid group. However, the Bestvina--Handel algorithm quickly becomes computationally intractable for large braid words, and matrix methods give only lower bounds, which are often poor for large braids. Our method is computationally fast and appears to give exponential convergence towards the exact entropy. As an illustration we apply our approach to the braiding of both periodic and aperiodic trajectories in the sine flow. The efficiency of the method allows us to explore how much extra information about flow entropy is encoded in the braid as the number of trajectories becomes large.

Extension of a chaos control method to unstable trajectories on infinite- or finite-time intervals: Experimental verification

In experiments for single and coupled pendula, we demonstrate the effectiveness of a new control method based on dynamical systems theory for stabilizing unstable aperiodic trajectories defined on infinite- or finite-time intervals. The basic idea of the method is similar to that of the OGY method, which is a well-known, chaos control method. Extended concepts of the stable and unstable manifolds of hyperbolic trajectories are used here.

Complex Dynamics in a Hysteretic Relay Feedback System with Delay

Relay feedback systems are used to control engineering devices. In practice the switching between different functional forms of the system is never instantaneous, but takes place after a small delay. In this paper we analyse the dynamics and bifurcations of a representative example of such systems. In the absence of delay, negative feedback results only in unimodal symmetric limit cycles, but positive feedback can lead to aperiodic trajectories and chaos. In the presence of delay, the system can behave as an equivalent system without delay, provided that the delay is small in a sense which we define precisely. For larger delays, we identify a new bifurcation phenomenon, an event collision, where the delayed switching manifold intersects the relay hysteretic lines. In this case the dynamics become much more complicated.

Periodicity in the transient regime of exhaustive polling systems

We consider an exhaustive polling system with three nodes in its transient regime under a switching rule of generalized greedy type. We show that, for the system with Poisson arrivals and service times with finite second moment, the sequence of nodes visited by the server is eventually periodic almost surely. To do this, we construct a dynamical system, the triangle process, which we show has eventually periodic trajectories for almost all sets of parameters and in this case we show that the stochastic trajectories follow the deterministic ones a.s. We also show there are infinitely many sets of parameters where the triangle process has aperiodic trajectories and in such cases trajectories of the stochastic model are aperiodic with positive probability.

Normal-Mode Analysis of a Baroclinic Wave-Mean Oscillation

Abstract The stability of a time-periodic baroclinic wave-mean oscillation in a high-dimensional two-layer quasigeostrophic spectral model is examined by computing a full set of time-dependent normal modes (Floquet vectors) for the oscillation. The model has 72 × 62 horizontal resolution and there are 8928 Floquet vectors in the complete set. The Floquet vectors fall into two classes that have direct physical interpretations: wave-dynamical (WD) modes and damped-advective (DA) modes. The WD modes (which include two neutral modes related to continuous symmetries of the underlying system) have large scales and can efficiently exchange energy and vorticity with the basic flow; thus, the dynamics of the WD modes reflects the dynamics of the wave-mean oscillation. These modes are analogous to the normal modes of steady parallel flow. On the other hand, the DA modes have fine scales and dynamics that reduce, to first order, to damped advection of the potential vorticity by the basic flow. While individual WD modes have immediate physical interpretations as discrete normal modes, the DA modes are best viewed, in sum, as a generalized solution to the damped advection problem. The asymptotic stability of the time-periodic basic flow is determined by a small number of discrete WD modes and, thus, the number of independent initial disturbances, which may destabilize the basic flow, is likewise small. Comparison of the Floquet exponent spectrum of the wave-mean oscillation to the Lyapunov exponent spectrum of a nearby aperiodic trajectory suggests that this result will still be obtained when the restriction to time periodicity is relaxed.

Topological Entropy of Braids on the Torus

We present a fast method for computing the topological entropy of braids on the torus. This work is motivated by the need to analyze large braids when studying two‐dimensional flows via the braiding of a large number of particle trajectories. Our approach is a generalization of Moussafir’s technique for braids on the disk. Previous methods for computing topological entropy include the Bestvina–Handel train‐track algorithm and matrix representations of the braid group. However, the Bestvina–Handel algorithm is computationally intractable for large braid words, and matrix methods give only lower bounds, which are often poor for large braids. Our method is computationally fast and gives exponential convergence towards the exact entropy. As an illustration we apply our approach to the braiding of both periodic and aperiodic trajectories in the sine flow.

Chaos in Bohmian quantum mechanics

In our recently published paper 'Chaos in Bohmian quantum mechanics' we criticized a paper by Parmenter and Valentine (1995 Phys. Lett. A 201 1), because the authors made an incorrect calculation of the Lyapunov exponent in the case of Bohmian orbits in a quantum system of two uncoupled harmonic oscillators.After our paper was published, we became aware of an erratum published by the same authors (Parmenter and Valentine 1996 Phys. Lett. A 213 319) that recognized the error made in their previous calculations. The authors realized that, when correctly calculated, 'aperiodic trajectories with well defined boundaries...have vanishing Lyapunov exponents', i.e., they are not chaotic.We want to supplement our paper with a reference to this erratum. The generic calculation of Lyapunov exponents in Bohmian quantum systems remains an original contribution of our paper (section 2).

On the constructive role of noise in stabilizing itinerant trajectories in chaotic dynamical systems.

This work aims at studying dynamical models of neural networks, which exhibit phase transitions between states of various complexities. We use the biologically motivated KIII model, which has demonstrated excellent performance as a robust dynamical memory device. KIII is a high-dimensional dynamical system with extremely fragmented boundaries between limit cycles, tori, fixed points, and chaotic attractors. We study the role of additive noise in the development of itinerant trajectories. Noise not only stabilizes aperiodic trajectories, but there is an optimum noise level with highly itinerant behavior. We speculate that the previously found optimum classification performance of KIII as a function of the noise level, also identified as chaotic resonance, is related to chaotic itinerant oscillations among various ordered states.

Observation of mixed-mode oscillations in spin-wave experiments

High-power ferromagnetic resonance experiments in a 1-mm-diam yttrium iron garnet sphere, driven at 8.9 GHz at room temperature, reveal evidence for a universal scenario not yet observed in this system, namely, the periodic–chaotic sequence. Well above the first-order Suhl instability threshold (P/PC∼6 dB), low-frequency (f∼200 kHz) auto-oscillations consisting of large-amplitude peaks followed by n small undulations (mixed-mode oscillations) have been observed. Both periodic and aperiodic trajectories were detected with n=1, 2, 3, and 4, which are reminiscent of the so-called homoclinic chaos. A discussion is made on the basis of a standard two-mode model.