Showing papers in "Chaos Solitons & Fractals in 1997"

Synchronization of two Lorenz systems using active control

Abstract: Using techniques from active control theory, we demonstrate that a coupled Lorenz system can be synchronized The synchronization is verified using the Simulink feature in MATLAB

Power-law sensitivity to initial conditions—New entropic representation

Abstract: The exponential sensitivity to the initial conditions of chaotic systems (e.g. D = 1) is characterized by the Liapounov exponent λ, which is, for a large class of systems, known to equal the Kolmogorov-Sinai entropy K. We unify this type of sensitivity with a weaker, herein exhibited, power-law one through (for a dynamical variable x) lim Δ x(0)→0 [ Δ x (t)] [ Δ x (0)] = [1 + (1 − q)λ q t] 1 (1 − q) (equal to eλ1t for q = 1, and proportional, for large t, to t 1 (1 − q) for q ≠ 1;. We show that gl (∀q), where Kq is the generalization of K within the non-extensive thermostatistics based upon the generalized entropic form S q ≡ (1 − ∑ i p i q ) (q − 1) ( hence , S 1 = −Σ i p i ln p i ) . The well-known theorem λ1 = K1 (Pesin equality) is thus extended to arbitrary q. We discuss the logistic map at its threshold to chaos, at period doubling bifurcations and at tangent bifurcations, and find q ≈ 0.2445, q = 5 3 and q = 3 2 , respectively. 05.45. + b; 05.20. − y; 05.90. + m.

Dynamics of oscillators with impact and friction

Abstract: In the present paper two types of nonsmooth oscillators are investigated: an impact oscillator and a self-sustained friction oscillator. Both are nonsmooth one degree of freedom oscillators with harmonic external excitation. Here the different types of motion, bifurcation diagrams and Poincare maps are determined from experiments. These results will be compared with numerical results on the basis of the identified impact and friction models. The nonsmooth third-order systems show rich bifurcational behaviour which is analysed by numerical simulations but also using mapping approaches. Two different formalisms for the calculation of the Lyapunov exponents are applied. The latter one requires special considerations in the given case of nonsmooth systems. Furthermore, the embedding dimension is gained applying the method of false nearest neighbours. In the case of coexisting solutions further analysis is done by means of bifurcation and stability analysis and the cell-mapping approach.

Multifractal scaling laws in the breaking behaviour of disordered materials

Abstract: A strong relationship between critical phenomena and fractals can be recognized, implying a deep analogy between geometry and physics: the common underlying rule is the scale invariance, which forms the basis of the renormalization group theory, successfully applied to the analysis of phase transitions. In the case of fracture of heterogeneous media, the macroscopic effect of the disordered microstructure turns into the incomplete self-similarity of the phenomenon with respect to the main macroscopical quantities, which have to be renormalized and assume non-integer (anomalous) dimensions. The dimensional fractional increment of the dissipation space in the case of fracture energy, as well as the dimensional fractional decrement of the material ligament in the case of ultimate strength, are shown to explain the experimental size-dependent trends of these mechanical quantities. On the other hand, the continuous vanishing of fractality with increasing the observation scale seems to be peculiar of all natural fractals (multifractals). Extrapolating to physics, this implies that the effect of microstructural disorder on the mechanical behaviour becomes progressively less important for larger structures (i.e. large when compared with the microstructural characteristic size) where the disordered microstructure is somehow homogenized. Therefore, the scale effect should vanish in the limit of structural size tending to infinity, where asymptotic values of the physical quantities can be determined. Two multifractal scaling laws are proposed, for fracture energy and tensile strength, respectively, where the dimensional transition (with increasing structural size) from a microscopic Brownian disorder, to a macroscopic Euclidean behaviour, is controlled by the slope in the bilogarithmic diagram. Best-fitting of relevant experimental results has confirmed the soundness of this new approach.

Fractal effects of surface roughness on the mechanical behavior of rock joints

Abstract: Extensive studies show that the naturally developed rock joint surfaces have the properties of fractals. The surface roughness of rock joints can be well described within the framework of fractal geometry. To give a better understanding of the roughness of rock fracture surfaces in relation to the mechanical properties and behavior of rock joints in loading, a systematic investigation has been carried out. By means of a laser scanning instrument, the fracture surfaces induced in rocks are measured. The fractal characters of rough fracture surfaces of rocks are analysed according to the variogram method, which elaborates explicitly the importance of the geometric parameters — fractal dimension D and the intercept A . Investigation extends to the anisotropy and heterogeneity of rock fracture surfaces, and the scale effect on the fractal estimation. The shear tests on rock joints show a combined effect of the fractal parameters on the mechanical properties and behavior of rock joints. To control the roughness and show the effects in a direct manner, a series of fractal joints with different fractal dimensions are generated on the basis of the Weierstrass-Mandelbrot function and then manufactured in the polycarbonate plates. By using the photoelastic method, the manners of normal and shear deformation, the stress field in the vicinity of the joint surface and the contact behaviour of the fractal joints have been studied in detail under uniaxial compression and direct shear. Based on the experimental results, the dependence of deformation stiffness on the fractal dimensions of rock joints has been recognized. The empirical criteria of shear strength and the evolution law of surface damage during the shear process are developed.

A linear continuous feedback control of Chua's circuit

Abstract: A continuous feedback controller is designed to suppress and eliminate the chaotic behavior of Chua's circuit. This controller is constructed by the following two phases. First, the set of equilibrium points is extended by virtue of embedding suitable feedback terms in the control channel. Second, a state error term is then added to steer the dynamics of the control system to a fixed point or a limit cycle. The stability region of Chua's circuit with control is determined via the Routh-Hurwitz criterion. Some numerical simulations are given to demonstrate the effect of the control design.

Why chaos is rarely observed in natural populations

Abstract: An attempt has been made to understand why chaotic dynamics have received poor evidential support from field studies. Our study opens up the possibility that the cause of failure might not be poor quality of data, as pointed out by earlier authors, but an ecological reality. We have designed two model food chains to examine whether there is a biological basis for the crisis. This investigation is effected with the help of a new method which we introduce at an appropriate place in the text. The fact that chaos exists in a narrow range of parametric values in both the model systems suggests that the crisis indeed has a biological origin.

Generalized Floquet theory for stationary Schrödinger operators in one dimension

Abstract: Floquet theory for ordinary differential operators with periodic coefficients is well known and fundamental to study the stability of solutions and spectral properties of the operators. Generalization of Floquet theory to differential operators with almost periodic or more random coefficients have been done by several authors, by applying C ∗ -algebra theory, dynamical theory and probability theory. The purpose of this paper is to give a general point of view of Floquet theory using the Weyl's functions systematically, which was initiated by Johnson-Moser.

Variable structure control approach for controlling chaos

Abstract: A chaotic system can be considered as a system that has its own limited energy source. A smart way of controlling chaos is to use the energy source and redirect it towards the region where desired performance is expected. A small perturbation of a parameter in the chaotic systems can serve the purpose. Variable structure control strategy as a natural candidate is examined in this paper for control of chaos in dynamical systems. It is shown that, by switching between two values of a perturbed parameter in chaotic systems, a sliding region can be created in which the desired performance is achievable. The stabilization and tracking of a periodic signal of the Rossler system are studied as an example. The effectiveness of the approach is analyzed and demonstrated with computer simulations.

On some controllability conditions for chaotic dynamics control

Abstract: In this paper, we present some conventional feedback controller design principles for chaos control, with mathematical controllability conditions derived via the Lyapunov function methods. The chaotic Chua's circuit and Duffing oscillator are used as examples to illustrate the fundamental concepts and basic methodology employed by this unified Lyapunov approach, in both linear and non-linear controllers design, for the control of chaotic dynamics.

Fractal functions and their applications

Abstract: This paper reviews how elements from the theory of fractal functions are employed to construct scaling vectors and multiwavelets. Emphasis is placed on the one-dimensional case, however extensions to IR{sup m} are indicated.

Chaotic and stochastic dynamics of orthogonal metal cutting

Abstract: A model of the cutting process is formulated based on a random variation of the specific cutting resistance. The cutting resistance is generated as a one-dimensional univariate Gaussian process using the spectral representation method. The random responses are discussed and compared with the deterministic ones within the ranges of parameters where chaotic motion occurs. Contrary to a claim for continuous systems, in which the variance of the noise dampens the chaotic vibration, such a behavior is not observed in the discontinuous system. The chaotic response is still present and the stochasticity induces immense impact forces during the transient period.

Control of chaos by nonfeedback methods in a simple electronic circuit system and the FitzHugh-Nagumo equation

Abstract: Various control algorithms have been proposed in recent years to control chaotic systems. These methods are broadly classified into feedback and nonfeedback methods. In this paper, we make a critical analysis of nonfeedback methods such as (i) addition of constant bias, (ii) addition of second periodic force, (iii) addition of weak periodic pulse, and (iv) entrainment control. We apply these methods to a simple electronic circuit, namely, the Murali-Lakshmanan-Chua circuit system and FitzHugh-Nagumo equation. We make a comparative study of the various features associated with the algorithms.

An adaptive method for the approximation of the generalized cell mapping

Abstract: For the prediction of the long-term behavior and global analysis of nonlinear dynamical systems the generalized cell mapping is an efficient and powerful numerical tool. Since in general the generalized cell mapping cannot be computed directly it is necessary to develop a numerical method which allows us to approximate it. The purpose of our paper is to present such a method, where the approximation works in an adaptive way. Our method has the advantage that it can be controlled by a deterministic a priori error estimate. We illustrate its efficiency by an example.

Remarks on superstrings, fractal gravity, Nagasawa's diffusion and Cantorian spacetime

Abstract: This paper discusses possible connections between quite different and partly rather remote disciplines such as hyper-dimensional sphere-packing lattices, superstring theory and the Cantorianfractal spacetime approach to particle physics. Furthermore, relations to fractional diffusion, Nagasawa's diffusion, quantum chaos and fractal gravity are briefly considered.

Localized coherent structures and integrability in a generalized (2 + 1)-dimensional nonlinear Schrödinger equation

Abstract: A generalized (2 + 1)-dimensional nonlinear Schrodinger equation introduced recently by Fokas is investigated and is shown to admit the Painleve property. The Hirota bilinearization directly follows from the singularity analysis. Localized dromion solutions, which arise essentially due to the interaction of two nonparallel ghost solitons and localized breather solutions (time oscillating solutions), are constructed using the Hirota method. This method can be rigorously pursued to generate multidromions and multibreathers.

Two new classes of exact solutions for the KdV equation via Bäcklund transformations

Abstract: The Korteweg-de Vries equation which includes nonlinear and dispersive terms quadratic in the wave amplitude is considered. The exact solutions can be obtained by the AKNS class. The technique developed relies on the construction of the wave functions which are solutions of the associated AKNS system; that is, a linear eigenvalue problem in the form of a system of first order partial differential equations. The method of characteristics is used and Backlund transformations (BTs) are employed to generate two new solutions from the old one.

An experimental impact oscillator

Abstract: The present work considers the impacting behavior of a piecewise linear experimental system, a previous version of which has successfully displayed a wide variety of non-linear, including chaotic, behavior in other previous experiments [1–3]; making use of the ‘ball rolling on a hill’ concept, a cart is constrained to roll along an ‘energy well’ track, shaped in the form of a parabola such that the governing equations of motion of the cart are almost linear. In contrast to previous studies, where the impact condition was fixed (usually at equilibrium), the rigid barrier is placed at varying positions along the track in this study to provide a displacement constraint that gives rise to a discontinuity in the restoring force on the cart (provided solely by gravity in the non-impacting region). Varying the position of the impact can significantly change the overall behavior of the system, and this experimental study will concentrate on behavior where the impact point is varied among three different positions along the parabolic track, one chosen at a position less than equilibrium, one chosen at equilibrium, and one chosen greater than equilibrium; wide and varied responses are observed for these cases. Data were collected with the LabVIEW object-oriented, programmable interface, about which a few salient features are also discussed.

Safety analysis of ship rolling in rough sea

Abstract: Non-linear ship rolling and capsizing in irregular waves is analysed by a single degree-of-freedom system. The random wave excitation depends on sea state, ship speed and ship direction to wave propagation. The governing differential equation of motion is integrated by applying the harmonic acceleration method. Ship response is presented in time and frequency domains. A safe basin in the initial value plane is constructed for different values of the three excitation parameters mentioned above in order to determine the probability of ship survival, which is presented in a PC radar map as a function of heading angle.

Lax pairs, symmetries and conservation laws of a differential-difference equation—Sato's approach

Abstract: Based on the known elementary introduction of Sato theory for differential equations, the differential-difference equation which belongs to the single component KP family has been considered in the framework of Sato theory. We show that in this natural framework, Lax pairs, symmetries and conservation laws can be obtained in a systematic way.

Fractal gravity and symmetry breaking in a hierachical Cantorian space

Abstract: In this paper we determine the exact expectation value and standard deviation for the dimensionality of a Cantorian spacetime. We show further that the symmetry of the universe is a precondition and the origin of its asymmetry. Connections to time symmetry breaking and the generalization of complex numbers are considered. Finally, a link between general relativity and Cantorian spacetime is proposed which amounts to the claim that the Bethe lattice-like fractalization of microspace is the origin of gravity.

Singularity structure analysis and Hirota's bilinearisation of the coupled integrable dispersionless equations

Abstract: We investigate the singularity structure analysis of coupled integrable dispersionless equations and we show that the system possesses the Painleve property The associated Backlund transformation is constructed and Hirota's bilinearisation is also obtained through dependent variable transformations Finally, we reduce the system to the well-known sine-Gordon equation

Boundary value problems and brownian motion on fractals

Abstract: A physical state in a domain is often described by a model containing a linear partial differential equation and associated boundary conditions. The mathematical tools required to study this are well known if the boundary of the domain is smooth enough or if the boundary is smooth except for one or several corners. But in reality the boundary of the domain is usually not smooth. The typical situation is rather that the boundary is strongly broken with an intricate detailed structure and maybe that the boundary exhibits similar patterns in different scales. This means that the boundary is typically a fractal showing some kind of self-similarity: a magnification of a part of the boundary has, in some sense, the same structure as the whole boundary. A typical example of a domain in the plane having a boundary of this kind is von Koch's snowflake domain. In the case of a fractal boundary the classical tools and theorems no longer hold. How does one provide the mathematical background in this case? This is the main topic of this survey paper. However, we also study Brownian motion on fractals.

Classical particles and the Dirac equation with an electromagnetic field

Abstract: The Einstein-Kac lattice model of Brownian motion in one dimension is extended to include massless particles which do not themselves scatter, but which facilitate scattering of Brownian particles. The model is completely classical and no formal quantization is employed. However by observing second-order effects in the distribution of Brownian particles we recover directly the components of Dirac wave functions. Furthermore the probability densities describing the massless particles are shown to form ‘fields’ which obey Maxwell's equations. This result extends the context of the Dirac equation to include the physics of ensembles of classical particles with contact interactions. In this context all ‘quantum objects’, including the wave function itself, are observable in terms of distributions of particles on a lattice.

Painlevé analysis and Bäcklund transformation for a three-dimensional Kadomtsev-Petviashvili equation

Abstract: In this paper, we have analysed the integrability properties of a three-dimensional Kadomtsev-Petviashvili equation by Painleve test and the associated Backlund transformation is constructed straightforwardly from the Painleve analysis.

Estimation of Lyapunov exponents of ECG time series—The influence of parameters

Abstract: The influence of working parameters on the estimation of Lyapunov spectra from limited data sets is investigated, using Lorenz, Rossler and Henon attractors, and an empirical law is presented. Application to electrocardiograms from the MIT-BIH Arrhythmias Database shows a smaller divergence rate (λ1) and convergence rate ( ¦ λ 3 ¦ or ¦ λ 4 ¦ ) in pathological than normal data.

On El Naschie's complex time and gravitation

Abstract: El Naschie proposed in Chaos, Solitons & Fractals , 5 , 1551–1555, 1995 [1], a complex time measure and demonstrated in this connection that massless particles may, under such conditions, achieve an infinite velocity whilst adhering to all prerequisites of the special theory of relativity. The aim of the present paper is to generalize this result in the presence of a relativistic gravitational field. The method is based on Einstein's equivalence principle whereby the action of a gravitational field on a test particle is locally compensated by a suitable choice of the acceleration of the particle.

Quantum chaos in open versus closed quantum dots: Signatures of interacting particles

Abstract: This paper reviews recent studies of mesoscopic fluctuations in transport through ballistic quantum dots, emphasizing differences between conduction through open dots and tunneling through nearly isolated dots. Both the open dots and the tunnel-contacted dots show random, repeatable conductance fluctuations with universal statistical properties that are accurately characterized by a variety of theoretical models including random matrix theory, semiclassical methods and nonlinear sigma model calculations. We apply these results in open dots to extract the dephasing rate of electrons within the dot. In the tunneling regime, electron interaction dominates transport since the tunneling of a single electron onto a small dot may be sufficiently energetically costly (due to the small capacitance) that conduction is suppressed altogether. How interactions combine with quantum interference are best seen in this regime.

Quantum transport in open mesoscopic cavities

Abstract: In this review we describe the results of magneto-transport studies in open quantum dots, in which electronic motion is expected to be predominantly ballistic in nature. The devices themselves are realized in different semiconductor materials, using quite distinct fabrication techniques. Electron interference is an important process in determining the electrical properties of the devices at low temperatures and is manifested through the observation of periodic magneto-conductance fluctuations. These are found to result from selective excitation of discrete cavity eigenstates by incoming electrons, which are directed into a collimated beam by the input point contact. Under conditions of such restricted injection, quantum mechanical simulations reveal highly characteristic wavefunction scarring, associated with the remnants of a few classical orbits. The scarring is built up by interference between electrons, confined within the cavities over very long time scales, suggesting the underlying orbits are highly stable in nature. This characteristic is also confirmed by the results of experiment, which reveal the discrete components dominating the interference to be insensitive to changes in lead opening or temperature. The fluctuations decay with increasing temperature, although they can nonetheless still be resolved at a few degrees kelvin. This characteristic is confirmed by independent studies of devices, fabricated using very different techniques, further demonstrating the universal nature of the behavior we discuss here. These results therefore demonstrate that the correct description of electron interference in open quantum cavities, is one in which only a few discrete orbits are excited by the collimating action of the input lead, giving rise to striking wavefunction scarring with measurable magneto-transport results.