Dimension Of Chaotic Attractor Research Articles

On the Existence of Low-Dimensional Chaos of the Tropical Cyclone Intensity in an Idealized Axisymmetric Simulation

Abstract This study examines the potential limit in predicting tropical cyclone (TC) intensity under idealized conditions. Using the phase-space reconstruction method for TC intensity time series obtained from the CM1 idealized simulations, it is found that CM1 axisymmetric dynamics contain low-dimensional chaos at the maximum intensity equilibrium. Examination of several attractor invariants including the largest Lyapunov exponent, the Sugihara–May correlation, and the correlation dimension captures a consistent range of the chaotic attractor dimension between 4 and 5 for TC intensity at the maximum intensity equilibrium. In addition, the intensity error doubling time estimated from the largest Lyapunov exponent is roughly 1–3 h, which accords with the decay time obtained from the Sugihara–May correlation. Furthermore, the findings in this study reveal a relatively short TC intensity predictability limit for CM1, which is ∼3–9 h based on the maximum tangential wind but noticeably longer for the minimum central pressure (∼12–18 h) after reaching the mature stage. So long as the traditional metrics for TC intensity such as the maximum surface wind or the minimum central pressure is used for intensity forecast, our results support that TC intensity forecast errors will not be reduced indefinitely in any model, even in the absence of all model and observational errors. As such, the future improvement of TC intensity forecast should be based on different metrics beyond the absolute intensity errors that are currently used in real-time intensity verification. Significance Statement Using the phase-space reconstruction method for tropical cyclone (TC) intensity time series obtained from idealized axisymmetric simulations, we show that TC axisymmetric dynamics in CM1 possesses low-dimensional chaos at the maximum intensity equilibrium. This low-dimensional dynamics explains the long tradition of representing TC intensity by a few measures as in the current practice. The chaotic property of CM1 axisymmetric dynamics also suggests a relatively short predictability range for TC intensity at the maximum intensity equilibrium. The potential existence of low-dimensional chaos for TC intensity in CM1 idealized simulations as found in this study supports the use of different intensity verification metrics beyond the traditional absolute intensity errors currently used in operational model evaluation.

Hausdorff dimension of chaotic attractors in a class of nonsmooth systems

Fractal dimension is an important feature of a chaotic attractor. Generally, the rigorous value of Hausdorff dimension of a chaotic attractor is not easy to compute. In this work, we consider a class of discontinuous piecewise linear maps. Initially, we determine a set of parameter values in which the maps have a chaotic attractor with an Sinai-Ruelle-Bowen measure. Then we give a lower bound and an upper bound of the Hausdorff dimension of the attractor. Our rigorous analysis shows that the two bounds are equal, and thus the exact formula of the Hausdorff dimension of the attractor is obtained. Moreover, the relationship between the Hausdorff dimension and the parameter values is discussed in terms of the derived formula.

Control and synchronization of fractional‐order chaotic satellite systems using feedback and adaptive control techniques

SummaryIn this article, we present the basic concepts of fractional calculus and control and synchronization of fractional‐order chaotic satellite system . Existence and uniqueness solutions of fractional‐order satellite system are discussed and local stability of the system at the equilibrium points are studied. The lowest dimension of chaotic attractor of satellite system is 2.88 which is obtained through utilization of the fractional dynamics and computational simulation. Lyapunov exponents and bifurcation diagrams are drawn to measure the existence of chaos in the system. Feedback control method for chaos control in the fractional‐order satellite system is achieved. Synchronization of two identical noninteger order chaotic satellite systems are achieved through adaptive control methodology.

Complex Structure in Modified Henon Map and Its Dynamic Evolution

Strange evolutionary behaviours of modified discrete Henon map investigated in the framework of non-linear dynamics. The study carried out to visualise regular and chaotic evolutions and to study the complexity in the system. Bifurcation diagrams obtained by varying a parameter while keeping other parameter fixed. Regular and chaotic attractors are drawn for different sets of parameters. During this process a multifold strange attractor also emerged for certain pair of parameters. Numerical calculations performed to calculate Lyapunov exponents for regular and chaotic cases. In order to measure complexity in the system calculations further extended to obtain the topological entropies for sets of parameter values. Finally, correlation dimensions of chaotic attractors obtained with proper description of numerical setup. Results obtained are significant and presented through graphics.

Evaluating the predictability of central Indian rainfall on short and long timescales using theory of nonlinear dynamics

Abstract The theoretical and practical understanding of projected changes in rainfall is desirable for planning and adapting to climate change. In this study, finite size Lyapunov exponents (FSLE) are used to study error growth rates of the system at different timescales. This is done to quantify the impact of enhanced anthropogenic greenhouse gas emissions on the predictability of fast and slow varying components of central Indian rainfall (CIR). The CIR time series for this purpose is constructed using the daily gridded high-resolution India Meteorological Department (IMD) dataset and Coupled Model Inter-comparison Project phase 5 (CMIP5) output for historical run and three representative concentration pathways (RCP2.6, RCP4.5, and RCP8.5) from the HadGEM2-ES, IPSL-CM5A-LR, CCSM4, BCC-CSM1.1, and MPI-ESM-LR models. The analyzed CIR dataset reveals a low dimensional chaotic attractor, suggesting that CIR requires a minimum of 5 and maximum of 11 variables to describe the state of the system. FSLE analysis shows a rapid decrease in the Lyapunov exponent with increasing timescales. This analysis suggests a predictability of about 2–3 weeks for fast varying components at short timescale of the CIR and about 5–9 years for slow varying components at long timescales.

Comparison of Variations of TEC at Disturbed and Quiet Time Using Nonlinear Dynamics Tools

AbstractWe analyze the underlying dynamics at three different latitudes of ionosphere through total electron content‐series for 4 months. The global dynamics reconstructed from the data is shown to be deterministic with low dimensional chaotic attractors at all the locations, and the dynamics is seen to be less complex at the midlatitude station with smaller dimension of correlation (CD) and maximum Lyapunov exponent (MLE). The influence of geomagnetic storms on the ionospheric dynamics is studied, by analyzing the local dynamics during disturbed and quiet periods, reconstructed from segments of total electron content‐series of shorter duration. With positive MLEs and smaller CDs, the local dynamics is seen to be deterministic and chaotic at the low latitude, midlatitude, and high latitude during both the disturbed and quiet days. However, it is interesting to observe that the dynamics is less chaotic and complex during the disturbed days of the entire period of observation compared to the adjacent quiet days at the three locations. This is further evidenced by sample predictions made using nonlinear tools, yielding better predictions during storms. The smaller CD value during disturbed periods indicates that the geomagnetic activities tend to make a few of the variables dominant enough to control the essential dynamics of the system. Further, the MLEs of the disturbed days are found to increase nearly in a linear fashion from beginning to end of the observation period at all the locations considered. This is an indication of the strong influence of geomagnetic activities on the ionosphere and might have caused the restructuring of the ionospheric dynamics.

The Shortest Synchronization Time with Optimal Fractional Order Value Using a Novel Chaotic Attractor Based on Secure Communication

In this study, a novel three dimensional autonomous chaotic attractor was found and secure communication masking application was performed with optimal fractional order, which offers more precise and faster results than first order chaotic equations, via Pecaro Carroll synchronization algorithm. The shortest synchronization time was investigated with optimal fractional order value. In the novel secure communication synchronization application with fractional order chaotic system, there is an angle of 45° between the signals sent and received, which clearly shows that the system can be employed in secure communication.

Is a hyperchaotic attractor superposition of two multifractals?

In the context of chaotic dynamical systems with exponential divergence of nearby trajectories in phase space, hyperchaos is defined as a state where there is divergence or stretching in at least two directions during the evolution of the system. Hence the detection and characterization of a hyperchaotic attractor is usually done using the spectrum of Lyapunov Exponents (LEs) that measure this rate of divergence along each direction. Though hyperchaos arise in different dynamical situations and find several practical applications, a proper understanding of the geometric structure of a hyperchaotic attractor still remains an unsolved problem. In this paper, we present strong numerical evidence to suggest that the geometric structure of a hyperchaotic attractor can be characterized using a multifractal spectrum with two superimposed components. In other words, apart from developing an extra positive LE, there is also a structural change as a chaotic attractor makes a transition to the hyperchaotic phase and the attractor changes from a simple multifractal to a dual multifractal, equivalent to two inter-mingled multifractals. We argue that a cross-over behavior in the scaling region for computing the correlation dimension is a manifestation of such a structure. In order to support this claim, we present an illustrative example of a synthetically generated set of points in the unit interval (a Cantor set with a variable iteration scheme) displaying dual multifractal spectrum. Our results are also used to develop a general scheme to generate both hyperchaotic as well as high dimensional chaotic attractors by coupling two low dimensional chaotic attractors and tuning a time scale parameter.

Some New Results for the Generalized Lorenz System

The bound of a chaotic dynamical system is important for chaos control, chaos synchronization, estimating the dimensions of chaotic attractors, and other engineering applications. It is a difficult work to get the bounds of a chaotic dynamical system. This paper further studies the bounds of the generalized Lorenz system (2) for $$a_{11} 0,\lambda _3 <0$$ based on Lyapunov stability theory. To the best of our knowledge, it is a very difficult task to study the bounds of the system (2) for $$a_{11} 0,\lambda _3 <0$$ and there are few papers to study the bounds of chaotic systems for this case.

Dynamical analysis and numerical simulation of bloch chaotic system

In this article, some dynamics of Bloch chaotic system have been studied. Based on Lagrange multiplier method, optimization theory, and the generalized positively definite and radially unbound Lyapunov functions with respect to the parameters of the system, we derive the ultimate bound and a family of mathematical expressions of globally exponentially attractive sets for this system with respect to the parameters of system. The results obtained in this article provides theory basis for chaotic synchronization, chaotic control, Hausdorff dimension, and Lyapunov dimension of chaotic attractors of Bloch chaotic system. © 2016 Wiley Periodicals, Inc. Complexity 21: 201–206, 2016

Characterization of intermittency at the onset of turbulence in the forced and damped nonlinear Schrödinger equation

In this paper we characterized intermittent transitions from temporal chaos to turbulence in the forced and damped nonlinear Schrödinger equation. We demonstrate using finite time Lyapunov exponents that during the transition a fraction of unstable periodic orbits embedded in a low dimensional chaotic attractor loses transversal stability, in a way that nearby trajectories are expelled away from its vicinity (a mechanism referred to as intermittency induced by Unstable Dimension Variability). During the transition, an appropriate decomposition of the Fourier phase space into transversal and longitudinal modes is performed. The analysis of modes dynamics sheds new light in the understanding of intermittency in spatially extended dynamical systems. Subsequently a perturbation is applied to the system in order to control the intermittent extreme events and reduce their occurrence.

Qualitative behavior of the Lorenz‐like chaotic system describing the flow between two concentric rotating spheres

In this article, the bounds of the Lorenz‐like chaotic system describing the flow between two concentric rotating spheres have been studied. Based on Lagrange multiplier method, the function extremum theory and the generalized positive definite and radially unbound Lyapunov functions with respect to the parameters of the system, we derive the ultimate bound and the globally exponentially attractive set for this system. The results that obtained in this article provides theory basis for chaotic synchronization, chaotic control, Hausdorff dimension and the Lyapunov dimension of chaotic attractors. © 2016 Wiley Periodicals, Inc. Complexity 21: 67–72, 2016

Nonlinear time series analysis of vibration data from a friction brake: SSA, PCA, and MFDFA

We use the methodology of singular spectrum analysis (SSA), principal component analysis (PCA), and multi-fractal detrended fluctuation analysis (MFDFA), for investigating characteristics of vibration time series data from a friction brake. SSA and PCA are used to study the long time-scale characteristics of the time series. MFDFA is applied for investigating all time scales up to the smallest recorded one. It turns out that the majority of the long time-scale dynamics, that is presumably dominated by the structural dynamics of the brake system, is dominated by very few active dimensions only and can well be understood in terms of low dimensional chaotic attractors. The multi-fractal analysis shows that the fast dynamical processes originating in the friction interface are in turn truly multi-scale in nature.

Bifurcations and chaos of a discrete mathematical model for respiratory process in bacterial culture

The discrete mathematical model for the respiratory process in bacterial culture obtained by Euler method is investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory, condition of existence of chaos in the sense of Marotto’s definition of chaos is proved. The bifurcation diagrams, Lyapunov exponents and phase portraits are given for different parameters of the model, and the fractal dimension of chaotic attractor was also calculated. The numerical simulation results confirm the theoretical analysis and also display the new and complex dynamical behaviors compared with the continuous model. In particular, we found that the new chaotic attractor, and new types of two or four coexisting chaotic attractors, and two coexisting invariant torus.

Turing patterns and long-time behavior in a three-species food-chain model

We consider a spatially explicit three-species food chain model, describing generalist top predator-specialist middle predator–prey dynamics. We investigate the long-time dynamics of the model and show the existence of a finite dimensional global attractor in the product space, L2(Ω). We perform linear stability analysis and show that the model exhibits the phenomenon of Turing instability, as well as diffusion induced chaos. Various Turing patterns such as stripe patterns, mesh patterns, spot patterns, labyrinth patterns and weaving patterns are obtained, via numerical simulations in 1d as well as in 2d. The Turing and non-Turing space, in terms of model parameters, is also explored. Finally, we use methods from nonlinear time series analysis to reconstruct a low dimensional chaotic attractor of the model, and estimate its fractal dimension. This provides a lower bound, for the fractal dimension of the attractor, of the spatially explicit model.

Controlling Unstable Chaos: Stabilizing Chimera States by Feedback

We present a control scheme that is able to find and stabilize an unstable chaotic regime in a system with a large number of interacting particles. This allows us to track a high dimensional chaotic attractor through a bifurcation where it loses its attractivity. Similar to classical delayed feedback control, the scheme is noninvasive, however only in an appropriately relaxed sense considering the chaotic regime as a statistical equilibrium displaying random fluctuations as a finite size effect. We demonstrate the control scheme for so-called chimera states, which are coherence-incoherence patterns in coupled oscillator systems. The control makes chimera states observable close to coherence, for small numbers of oscillators, and for random initial conditions.

THE GENERALIZED TIME-DELAYED HÉNON MAP: BIFURCATIONS AND DYNAMICS

We analyze the bifurcations of a family of time-delayed Hénon maps of increasing dimension and determine the regions where the motion is attracted to different dynamical states. As a function of parameters that govern nonlinearity and dissipation, boundaries that confine asymptotic periodic motion are determined analytically, and we examine their dependence on the dimension d. For large d these boundaries converge. In low dimensions both the period-doubling and quasiperiodic routes to chaos coexist in the parameter space, but for high dimensions the latter predominates and prior to the onset of chaos, the systems exhibit multistability. When the nonlinearity parameter is varied, the dimension of chaotic attractors in the systems changes smoothly with increasing number of non-negative Lyapunov exponents.

Relation between Topological Entropies and Fractal Dimensions of Chaotic Attractors of the Hénon Map

In two-dimensional chaotic dynamics, relationship between fractal dimensions and topological entropies is an important issue to understand the chaotic attractors of Hénon map. we proposed a efficient approach for the estimation of topological entropies through the study on the integral relationship between fractal dimensions and topological entropies. Our result found that there is an approximate linear relation between their topological entropies and fractal dimensions.

INFLUENCE OF SAMPLING LENGTH AND SAMPLING INTERVAL ON CALCULATING THE FRACTAL DIMENSION OF CHAOTIC ATTRACTORS

This paper intends to study the influences of the sampling length and sampling interval of time series on the chaotic attractors' fractal dimension calculation. Four kinds of univariate time-series signals from different chaotic systems were chosen, and then fractal dimensions of attractors under different sampling lengths and sampling intervals were calculated by the method of correlation dimension. The results show clearly that the chaotic attractors' fractal dimension is related to both the sampling length and the sampling interval. With the increase of the sampling length, all attractors' fractal dimensions tend to increase gradually first and then become stable. However, the fractal dimension remains stable only in a suitable range of the sampling interval, in which the attractor of the chaotic system can be reconstructed from one univariate time-series signal; if the sampling interval is unusually large or small, the fractal dimension will be unstable and the reconstructed attractor will be seriously distorted. Therefore, the dimension saturation method and the delay-coordinate's time difference method for determining the sampling length and the sampling interval were proposed separately, which are significant for improving the calculation accuracy for the chaotic attractor's dimension, reflecting the dynamics of complicated systems correctly, saving computational time as well as enhancing the computation efficiency.

Self-intersections of the Phase Trajectories as a Measure of the Embedding Dimension of Chaotic Attractors. Part I

Self-intersections of the Phase Trajectories as a Measure of the Embedding Dimension of Chaotic Attractors. Part I