Dimension Of Strange Attractor Research Articles

Globally Exponentially Attracting Sets and Heteroclinic Orbits Revealed

Motivated by the open problems on the global dynamics of the generalized four-dimensional Lorenz-like system, this paper deals with the existence of globally exponentially attracting sets and heteroclinic orbits by constructing a series of Lyapunov functions. Specifically, not only is a family of mathematical expressions of globally exponentially attracting sets derived, but the existence of a pair of orbits heteroclinic to S0 and S± is also proven with the aid of a Lyapunov function and the definitions of both the α-limit set and ω-limit set. Moreover, numerical examples are used to justify the theoretical analysis. Since the obtained results improve and complement the existing ones, they may provide support in chaos control, chaos synchronization, the Hausdorff and Lyapunov dimensions of strange attractors, etc.

System for ecological regulation of the biosynthesis of flavonoids as a strange attractor

Flavonoids are a large class of plant polyphenols with various adaptation functions to environmental factors. The biosynthesis of flavonoids is characterized by a high plasticity of their synthesis and multiple species differences. The HPLC spectra of flavonoid chromatograms from the populations of Juniperus sabina, Glycirrhiza korshinskyi, Achillea millefolia of the South Trans-Urals and pulations of Oxycoccus palustris, Chamaedaphne calyculata, Andromeda polifolia oligotrophic bogs in Western Siberia were studied. Each chromatograms of all species differed in the number of peaks, peak areas, and peak release times (compounds). It was shown that the system of flavonoid biosynthesis is fractal in nature. The species groups of flavonoids have properties of strange attractors. Using the Principal Component Analysis (PCA) clear differences between species and groups of species in contrasting ecosystems is shown. Thus, species-specific populations of flavonoids and compounds with similar physicochemical properties are a distinct regional product. In this regard, along with the search and selection of individual plant species based on the content of one compound, it seems appropriate to search for effective regional complexes of flavonoids

Quartic polynomial approximation for fluctuations of separation of trajectories in chaos and correlation dimension

We consider the cumulant generating function of the logarithm of the distance between two infinitesimally close trajectories of a chaotic system. Its long-time behavior is given by the generalized Lyapunov exponent providing the logarithmic growth rate of the kth moment of the distance. The Legendre transform of is a large deviations function that gives the probability of rare fluctuations where the logarithmic rate of change of the distance is much larger or much smaller than the mean rate defining the first Lyapunov exponent. The only non-trivial zero of is at minus the correlation dimension of the attractor which for incompressible flows reduces to the space dimension. We describe here general properties constraining the form of and the Gallavotti–Cohen type relations that hold when there is symmetry under time-reversal. This demands studying joint growth rates of infinitesimal distances and volumes. We demonstrate that quartic polynomial approximation for does not violate the Marcinkiewicz theorem on invalidity of polynomial form for the generating function. We propose that this quartic approximation will fit many experimental situations, not having the effective time-reversibility and the short correlation time properties of the quadratic Grassberger–Procaccia estimates. We take the existing for turbulent channel flow and demonstrate that the quartic fit is nearly perfect. The violation of time-reversibility for the Lagrangian trajectories of the incompressible Navier–Stokes (NS) turbulence below the viscous scale is considered. We demonstrate how the fit can be used for finding the correlation dimensions of strange attractors via easily measurable quantities. We provide a simple formula via the Lyapunov exponents, holding in quadratic approximation, and describe the construction of the quartic approximation. A different approximation scheme for finding the correlation dimension from expansion in the flow compressibility is also provided.

Predicting the bounds of large chaotic systems using low-dimensional manifolds.

Predicting extrema of chaotic systems in high-dimensional phase space remains a challenge. Methods, which give extrema that are valid in the long term, have thus far been restricted to models of only a few variables. Here, a method is presented which treats extrema of chaotic systems as belonging to discretised manifolds of low dimension (low-D) embedded in high-dimensional (high-D) phase space. As a central feature, the method exploits that strange attractor dimension is generally much smaller than parent system phase space dimension. This is important, since the computational cost associated with discretised manifolds depends exponentially on their dimension. Thus, systems that would otherwise be associated with tremendous computational challenges, can be tackled on a laptop. As a test, bounding manifolds are calculated for high-D modifications of the canonical Duffing system. Parameters can be set such that the bounding manifold displays harmonic behaviour even if the underlying system is chaotic. Thus, solving for one post-transient forcing cycle of the bounding manifold predicts the extrema of the underlying chaotic problem indefinitely.

Non-smooth saddle-node bifurcations II: Dimensions of strange attractors

We study the geometric and topological properties of strange non-chaotic attractors created in non-smooth saddle-node bifurcations of quasiperiodically forced interval maps. By interpreting the attractors as limit objects of the iterates of a continuous curve and controlling the geometry of the latter, we determine their Hausdorff and box-counting dimension and show that these take distinct values. Moreover, the same approach allows us to describe the topological structure of the attractors and to prove their minimality.

Self-Organization and Chaos in the Metabolism of Hemostasis in a Blood Vessel

A mathematical model of the metabolic process of formation of the hemostasis in a blood-carrying vessel is constructed. As distinct from the earlier developed model of the multienzyme prostacyclin-thromboxane system of blood, this model includes, for the first time, the influence of the level of “bad cholesterol”, i.e., low-density lipoproteins (LDLs), on the hemostasis. The conditions, under which the self-organization of the system appears, and the modes of autooscillations and chaos in the metabolic process, which affects the formation of hemostasis and the development of thrombophilia, are found. With the aid of a phase-parametric diagram, the scenario of their appearance is studied. The bifurcations of doubling of a period and the transition to chaotic oscillations as a result of the intermittence are found. The obtained strange attractors are formed due to a mixing funnel. The full spectra of Lyapunov’s exponents, KS-entropies, “predictability horizons”, and Lyapunov’s dimensions of strange attractors are calculated. The reasons for a change in the cyclicity of the given metabolic process, its stability, and the physiological manifestation in the blood-carrying system are discussed. The role of physiologically active substances in a decrease in the level of cholesterol in blood vessels is estimated.

Self-Organization and Fractality in the Metabolic Process of Glycolysis

Within a mathematical model, the metabolic process of glycolysis is studied. The general scheme of glycolysis is considered as a natural result of the biochemical evolution. By using the theory of dissipative structures, the conditions of self-organization of the given process are sought. The autocatalytic processes resulting in the conservation of cyclicity in the dynamics of the process are determined. The conditions of breaking of the synchronization of the process, increase in the multiplicity of a cyclicity, and appearance of chaotic modes are studied. The phase-parametric diagrams of a cascade of bifurcations, which characterize the transition to chaotic modes according to the Feigenbaum scenario and the intermittence, are constructed. The strange attractors formed as a result of the funnel effect are found. The complete spectra of Lyapunov indices and divergences for the obtained modes are calculated. The values of KS-entropy, horizons of predictability, and Lyapunov dimensions of strange attractors are determined. Some conclusions concerning the structural-functional connections in glycolysis and their influence on the stability of the metabolic process in a cell are presented.

Numerical Research on Self-Sustained Oscillation and Chaos of Natural Convection in Horizontal Fluids Layer

Numerical simulation was conducted to study self-sustained oscillation and chaos of natural convection in horizontal fluids layer heated from below. The numerical simulation was conducted using SIMPLE algorithm with a QUICK scheme. Discrete time series was analyzed to study stability problem. Correlation dimension was calculated, their frequency characteristics were analyzed and chaos attractor was extracted. The results showed that oscillation appeared when Ra was over 2×104 and chaos appeared when Ra was over 5×106. When parameter was varied, chaos appeared. Strange attractor was used to describe chaos system. Fractal dimension of strange attractor was 1.19. Saturated embedding dimension of phase space reconstruction was 5. The time delay was selected for 5.

Ulam method and fractal Weyl law for Perron-Frobenius operators

We use the Ulam method to study spectral properties of the Perron-Frobenius operators of dynamical maps in a chaotic regime. For maps with absorption we show that the spectrum is characterized by the fractal Weyl law recently established for nonunitary operators describing poles of quantum chaotic scattering with the Weyl exponent $\nu=d-1$, where $d$ is the fractal dimension of corresponding strange set of trajectories nonescaping in future times. In contrast, for dissipative maps we find the Weyl exponent $\nu=d/2$ where $d$ is the fractal dimension of strange attractor. The Weyl exponent can be also expressed via the relation $\nu=d_0/2$ where $d_0$ is the fractal dimension of the invariant sets. We also discuss the properties of eigenvalues and eigenvectors of such operators characterized by the fractal Weyl law.

Strange attractor in earthquake swarms near Valsad (Gujarat), India

Valsad district in south Gujarat near the western coast of the peninsular India experienced earthquake swarms since early February 1986. Seismic monitoring through a network of micro earthquake seismographs showed a well concentrated seismic activity over an area of 7 × 10 km2 with the depth of foci extending from 1 to 15 km. A total number of 21,830 earthquakes were recorded during March 1986 to June 1988. The daily frequency of earthquakes for this period was utilized to examine deterministic chaos through evaluation of dimension of strange attractor and Lyapunov exponent. The low dimension of 2.1 for the strange attractor and positive value of the largest Lyapunov exponent suggest chaotic dynamics in Valsad earthquake swarms with at least 3 parameters for earthquake predictability. The results indicate differences in the characteristics of deterministic chaos in intraplate and interplate regions of India.

Transition to chaos via the quasi-periodicity and characterization of attractors in confined Bénard-Marangoni convection

A study of dynamic regimes in Benard-Marangoni convection was carried out for various Prandtl and Marangoni numbers in small aspect ratio geometries (Γ = 2.2 and 2.8). Experiments in a small hexagonal vessel, for a large range of the Marangoni number (from 148 to 3636), were carried out. Fourier spectra and an auto-correlation function were used to recognize the various dynamic regimes. For given values of the Prandtl number (Pr = 440) and aspect ratio (Γ = 2.2), mono-periodic, bi-periodic and chaotic states were successively observed as the Marangoni number was increased. The correlation dimensions of strange attractors corresponding to the chaotic regimes were calculated. The dimensions were found to be larger than those obtained by other authors for Rayleigh-Benard convection in aspect ratio geometries of the same order. The transition from temporal chaos to spatio-temporal chaos was also observed. For Γ = 2.2, when larger values of the Marangoni number were imposed (Ma = 1581 for Pr = 160 and Ma = 740 for Pr = 440), spatial modes were involved through the convective pattern dynamics.

Bifurcation to chaos in the Bénard-Marangoni instability in a confined geometry

An experimental study of dynamical regimes in Bénard-Marangoni convection, for various Prandt and Marangoni numbers, has been carried out in a confined geometry. Indeed, a small hexagonal vessel allowing the formation of only one convective cell, for a large extent of the Marangoni number, has been used. Fourrier spectra and a correlation function have been used to recognize the various dynamical regimes. For fixed values of the Prantl number and aspect ratio, an oscillatory, a quasi-periodic and chaotic states, were successively observed, as the Marangoni number was increased. The correlation dimensions of strange attractors corresponding to the chaotic regimes were calculated. The dimensions were found to be larger then those calculated by other authors for the Rayleigh-Bénard convection in small aspect ratio geometries. The transition from temporal chaos to spatio - temporal chaos has also been observed. Indeed for higher values of the Marangoni number, spatial dynamics are observed.

Chaos in a double driven dissipative nonlinear oscillator.

We propose an anharmonic oscillator driven by two periodic forces of different frequencies as a time-dependent model for investigating quantum dissipative chaos. Our analysis is done in the framework of the statistical ensemble of quantum trajectories in a quantum state diffusion approach. The quantum dynamical manifestations of chaotic behavior, including the emergence of chaos, properties of strange attractors, and quantum entanglement, are studied by numerical simulation of the ensemble averaged Wigner function and von Neumann entropy.

Chaotic dynamic analysis of viscoelastic plates

The dynamic buckling of viscoelastic plates with large deflection is investigated in this paper by using chaotic and fractal theory. The material behavior is given in terms of the Boltzmann superposition principle. In order to obtain accurate computation results, the nonlinear integro-differential dynamic equation is changed into an autonomic four-dimensional dynamical system. The numerical time integrations of equations are performed by using the fourth-order Runge–Kutta method. And the Lyapunov exponent spectrum, the fractal dimension of strange attractors and the time evolution of deflection are obtained. The influence of geometry nonlinearity and viscoelastic parameter on the dynamic buckling of viscoelastic plates is discussed.

Chaotic regimes of antiferromagnetic resonance in a quasi-two-dimensional easy-axis antiferromagnet (NH3)2(CH2)4MnCl4

Chaotic regimes of the microwave energy absorption are observed experimentally and analyzed for two-dimensional metallorganic antiferromagnet (NH3)2(CH2)4MnCl4 at low temperatures under the conditions of nonlinear antiferromagnetic resonance. Relaxation oscillations of energy absorption are investigated in detail. Their frequency spectra, frequency–amplitude characteristics, and dependences of absorbed power on driving power and static magnetic field are studied. It is shown that the dynamics of relaxation oscillations undergoes a transition to chaos by “irregular periods.” Peculiarities of the transition are described consistently. Among other things, the conditions for the emergence of energy absorption regimes with a spike-like and a saw-tooth signal structure are determined, and the characteristics of chaotic oscillations such as the dimensions of strange attractors are calculated. The chaotic dynamics is found to be high-dimensional with a large contribution from noise which is of deterministic origin in the antiferromagnet under investigation.

Focal enhancement of chaotic strange attractor dimension in the left semantic (Wernicke) human cortex during reading without concomitant change in vigilance level

Non-averaged scalp-recorded brain potentials were analyzed in four right-handed humans while they were either silently reading a book, or in idle alert conditions. The dimensionality of chaotic strange attractors was estimated from the EEG at each of 14 scalp electrodes, using the point correlation dimension PD2 algorithm. When the subject was fully alert but not engaged in any specific task, PD2 was between 5.2 and 5.9. When the subject was silently reading a book, PD2 bifurcated to significantly higher values (6.6–6.9) in the left lateral anterior extrasylvian temporal cortex, but did not change in the adjacent left perisylvian cortex nor over the right side of the brain. The PD2 increases occurred without any significant change in vigilance level as assessed from the unchanged mean voltage of concomitant gamma (40 Hz) EEG oscillations. A small barely significant PD2 increase was only noticed in the left lateral precentral region. The increase of chaotic dimensionality in the left lateral extrasylvian temporal cortex during reading reflects enhanced processing operations in the semantic (Wernicke) areas and is in line with recent PET imaging data. These results provide for the first time evidence of a focal chaotic fractal dimension increase that is related to specific cognitive brain processing operations.

Strange attractor characteristics of earthquakes in Shillong Plateau and adjoining regions

Strange attractor dimensions were determined for two concentric areas of about 220 km (2°) and 440 km (4°) radii around WWSSN station at Shillong based on earthquake data during the period 1964 to 1992. Existence of deterministic chaos for both the areas was supported by the same fractal dimension of 6.1 and positive value of the largest Lyapunov exponent. The results have provided justification for delineating precursory seismicity patterns based on earthquakes up to epicentral distance of 440 km on different fault systems.Conspicuous higher value of the strange attractor dimension in Hindukush, Himalaya and northeast India than that in Koyna region in the peninsular India suggests a new criteria for distinction between interplate and intraplate earthquakes. Differences with subduction zone in Pacific vis‐à‐vis stress drop in great earthquakes (M≥8) and implications of the chaotic approach in earthquake hazard analysis have also been brought out.

Reservoir associated characteristics using deterministic chaos in Aswan, Nurek and Koyna reservoirs

Nurek, Aswan and Koyna reservoirs were affected by moderate earthquakes with continuing seismic activity. Microearthquake data recorded through local networks have been used to determine the strange attractor dimensions, using deterministic chaos which were found as 7,2, 3.8 and 4.8, respectively. This would imply that while 8 parameters are needed to model earthquakes near Nurek reservoir, only 4 to 5 parameters are needed for the Aswan and Koyna regions. The differences in the strange attractor dimension suggest them to be a measure of seismotectonics around such reservoirs.

Prediction of unspots using reconstructed chaotic system equations

Modeling of sunspots is important since they indicate the relative activity of the Sun which in turn influence different terrestrial properties. To predict the nonlinear and chaotic behavior of sunspot time series, the problem of reconstructing underlying system equations is studied. The proposed procedure for this is (1) based on the behavior of observed time series and dimension of strange attractor, find reference system equations (in our case the modified Rössler equations) that show similar basic features as the time series (e.g., appearance of attractor, amplitude and pseudoperiod), (2) assume a general structure of the governing system equations by Taylor series expansion, (3) use the reference equation systems as initial state in an updating procedure (extended Kalman filtering) to estimate the structure of the governing system equations. Using this procedure, results show that predictions on an average up to eight months ahead can be made with good agreement for sunspot time series after identifying the governing system equations. The extended Kalman filter was shown to be an efficient tool to identify parameter values and to make updated predictions of the chaotic system.

Strange attractor dimension as a new measure of seismotectonics around Koyna Reservoir, India : H. N. Srivastava, S. N. Bhattacharya & K. C. S. Ray, Earth & Planetary Science Letters, 124(1–4), 1994, pp 57–62.

Strange attractor dimension as a new measure of seismotectonics around Koyna Reservoir, India : H. N. Srivastava, S. N. Bhattacharya & K. C. S. Ray, Earth & Planetary Science Letters, 124(1–4), 1994, pp 57–62.