Showing papers in "Chaos Solitons & Fractals in 1996"
TL;DR: In this article, the authors introduced fractional derivatives of order α in time, with 0 for relaxation, diffusion, oscillations, and wave propagation, and showed that they are governed by simple differential equations of order 1 and 2 in time.
Abstract: The processes involving the basic phenomena of relaxation, diffusion, oscillations and wave propagation are of great relevance in physics; from a mathematical point of view they are known to be governed by simple differential equations of order 1 and 2 in time. The introduction of fractional derivatives of order α in time, with 0
925 citations
TL;DR: In this article, the authors modified the FitzHugh-Nagumo model of an excitable medium so that it describes adequately the dymanics of pulse propagation in the canine myocardium.
Abstract: We modified the FitzHugh-Nagumo model of an excitable medium so that it describes adequately the dymanics of pulse propagation in the canine myocardium. The modified model is simple enough to be used for intensive 3-dimensional (3D) computations of the whole heart. It simulates the pulse shape and the restitution property of the canine myocardium with good precision.
745 citations
TL;DR: In this paper, an integrable theory for perturbation equations engendered from small disturbances of solutions is developed, which includes various integrability properties of the perturbations, such as hereditary recursion operators, master symmetries, linear representations (Lax and zero curvature representations) and Hamiltonian structures, and provides us with a method of generating hereditary operators, Hamiltonian operators and symplectic operators starting from the known ones.
Abstract: An integrable theory is developed for the perturbation equations engendered from small disturbances of solutions. It includes various integrable properties of the perturbation equations, such as hereditary recursion operators, master symmetries, linear representations (Lax and zero curvature representations) and Hamiltonian structures, and provides us with a method of generating hereditary operators, Hamiltonian operators and symplectic operators starting from the known ones. The resulting perturbation equations give rise to a sort of integrable coupling of soliton equations. Two examples (MKdV hierarchy and KP equation) are carefully carried out.
342 citations
TL;DR: In this paper, the authors investigate the microeconomic foundations of Cournot duopoly games such that the reaction functions are unimodal and demonstrate that cost functions incorporating an interfirm externality lead to a system of coupled logistic equations.
Abstract: We are investigating microeconomic foundations of Cournot duopoly games such that the reaction functions are unimodal. We demonstrate that cost functions incorporating an interfirm externality lead to a system of coupled logistic equations. In the situation where agents take turns, we observe periodic and complex behavior. A closer analysis reveals some well-known local bifurcations. In a more general situation, where agents move simultaneously, we observe global bifurcations which typically occur in two-parameter families of two-dimensional endomorphisms.
240 citations
TL;DR: The scale covariant theory as mentioned in this paper is a new approach to the problem of the origin of fundamental scales and of scaling laws in physics, that consists of generalizing Einstein's principle of relativity (up to now applied to motion laws) to scale transformations.
Abstract: The theory of scale relativity is a new approach to the problem of the origin of fundamental scales and of scaling laws in physics, that consists of generalizing Einstein's principle of relativity (up to now applied to motion laws) to scale transformations. Namely, we redefine space-time resolutions as characterizing the state of scale of the reference system and require that the equations of physics keep their form under resolution transformations (i.e. be scale covariant). We recall in the present review paper how the development of the theory is intrinsically linked to the concept of fractal space-time, and how it allows one to recover quantum mechanics as mechanics on such a non-differentiable space-time, in which the Schrodinger equation is demonstrated as a geodesies equation. We recall that the standard quantum behavior is obtained, however, as a manifestation of a “Galilean” version of the theory, while the application of the principle of relativity to linear scale laws leads to the construction of a theory of special scale relativity, in which there appears impassable, minimal and maximal scales, invariant under dilations. The theory is then applied to its preferential domains of applications, namely very small and very large length- and time-scales, i.e. high energy physics, cosmology and chaotic systems.
174 citations
TL;DR: In this article, the authors present results of theoretical analysis, analogue simulations and numerical solutions of a particular but typical impacting system, and show that the transition between neighbouring periodic impact motions is never continuous, with the exception of singular points, where the existence boundaries and stability boundaries intersect.
Abstract: The motion of mechanical systems with impacts is strongly nonlinear. Many different types of periodic and chaotic impact motions exist even for simple systems with external periodic excitation forces. The group of fundamental periodic motions is characterized by the different number of impacts in one motion period, which equals the excitation force period. Every motion has a region in the space of system parameters in which the solution can exist and is stable. There exist transition regions, so-named hysteresis regions and beat motion regions, which lie between the zones of neighbouring fundamental impact motions. Transition regions are determined by the boundaries of existence which correspond to grazing bifurcations and by the boundaries of stability corresponding to the period-doubling and saddle-node bifurcations. The transition between neighbouring periodic impact motions is never continuous, with the exception of singular points, where the existence boundaries and stability boundaries intersect. Jump phenomena appear on the hysteresis region boundaries and subharmonic and chaotic motions exist in the transition beat motion region. This paper presents results of theoretical analysis, analogue simulations and numerical solutions of a particular but typical impacting system.
119 citations
TL;DR: In this article, the adjustment process by three Cournot oligopolists is studied in terms of an iso-elastic demand function and constant marginal costs, and the system can easily result in chaotic behaviour, and a muc...
Abstract: The adjustment process by three Cournot oligopolists is studied. An iso-elastic demand function and constant marginal costs are assumed. The system can easily result in chaotic behaviour, and a muc ...
108 citations
TL;DR: In this paper, the interval perturbation approximating formula is presented for evaluating interval eigenvalues of the interval matrix, which is a characteristic of the proposed method and illustrative numerical examples are provided.
Abstract: This paper deals with eigenvalue problems involving uncertain but non-random interval stiffness and/or mass matrices. If one views the deviation amplitude of the interval matrix as a perturbation around the nominal value of the interval matrix, one can solve the standard eigenvalue problem of the interval matrix by applying the interval extension to the matrix perturbation method. In this study, the interval perturbation approximating formula is presented for evaluating interval eigenvalues of the interval matrix. Inextensive computational effort is a characteristic of the proposed method. The illustrative numerical examples are provided.
105 citations
TL;DR: In this article, a regular approach to stability and bifurcation analysis in systems with mechanical collisions is proposed, based upon explicit formulas, expressing general solution matrices in terms of derivations of active forces as well as reaction ones.
Abstract: A regular approach to stability and bifurcation analysis in systems with mechanical collisions is proposed. It is based upon explicit formulas, expressing general solution matrices in terms of derivations of active forces as well as reaction ones. It is shown that the phenomenon of grazing impact, which was known to be a discontinuous bifurcation, can be regularized owing to the appropriate impact rule, which differs from the usual one. This results in a new classification of grazing bifurcations. In short, given periodic motion does not have to disappear: it might survive after such bifurcation and even preserve stability. A similar conclusion is valid with respect to bifurcation in systems with symmetry, though classifying conditions have another form. Mechanical examples are considered: linear oscillator with one or two stops and rigid block under periodic excitation.
80 citations
TL;DR: In this paper, it is shown that for the class of large non-integrable Poincare systems (LPS) the two descriptions are not equivalent, and that the trajectories of large LPS can be formulated in terms of trajectories or by statistical ensembles whose time evolution is described by the Liouville equation.
Abstract: Classical dynamics can be formulated in terms of trajectories or in terms of statistical ensembles whose time evolution is described by the Liouville equation It is shown that for the class of large non-integrable Poincare systems (LPS) the two descriptions are not equivalent Practically all dynamical systems studied in statistical mechanics belong to the class of LPS The basic step is the extension of the Liouville operator LH outside the Hilbert space to functions singular in their Fourier transforms This generalized function space plays an important role in statistical mechanics as functions of the Hamiltonian, and therefore equilibrium distribution functions belong to this class Physically, these functions correspond to situations characterized by ‘persistent interactions’ as realized in macroscopic physics Persistent interactions are introduced in contrast to ‘transient interactions’ studied in quantum mechanics by the S-matrix approach (asymptotically free in and out states) The eigenvalue problem for the Liouville operator LH is solved in this generalized function space for LPS We obtain a complex, irreducible spectral representation Complex means that the eigenvalues are complex numbers, whose imaginary part refers to the various irreversible processes such as relaxation times, diffusion etc Irreducible means that these representations cannot be implemented by trajectory theory As a result, the dynamical group of evolution splits into two semi-groups Moreover, the laws of classical dynamics take a new form as they have to be formulated on the statistical level They express ‘possibilities’ and no more ‘certitudes’ The reason for the new features is the appearance of new, non-Newtonian effects due to Poincare resonances The resonances couple dynamical events and lead to ‘collision operators’ (such as the Fokker-Planck operator) well-known from various phenomenological approaches to non-equilibrium physics These ‘collision operators’ represent diffusive processes and mark the breakdown of the deterministic description which was always associated with classical mechanics ‘Subdynamics’ as discussed in previous publications, is derived from the spectral representation The eigenfunctions of the Liouville operator have remarkable properties as they lead to long-range correlations due to resonances even if the interactions as included in the Hamiltonian are short-range (only equilibrium correlations remain short-range) This is in agreement with the results of non-equilibrium thermodynamics as the appearance of dissipative structures is connected to long-range correlations In agreement with previous results, it is shown that there exists an intertwining relation between LH and the collision operator Θ as defined in the text Both have the same eigenvalues and are connected by a non-unitary similitude ΛLHΛ−1 = Θ The various forms of Λ and their symmetry properties are discussed A consequence of the intertwining relation are ‘non-linear Lippmann-Schwinger’ equations which reduce to the classical linear Lippmann-Schwinger equations when the dissipative effects due to the Poincare resonances can be neglected Using the transformation operator Λ, we can define new distribution functions and new observables whose evolution equations take a specially simple form (they are ‘bloc diagonalized’) Dynamics is transformed in an infinite set of kinetic equations Starting with these equations, we can derive H -functions which present a monotonous time behavior and reach their minimum at equilibrium This requires no extra-dynamical assumptions (such as coarse graining, environment effects …) Moreover, our formulation is valid for strong coupling (beyond the so-called Van Hove's λ2t limit) We then study the conditions under which our new non-Newtonian effects are observable For a finite number N of particles and transient interactions (such as realized in the usual scattering experiments) we recover traditional trajectory theory To observe our new effects we need persistent interactions associated to singular distribution functions We have studied in detail two examples, both analytically and by computer simulations These examples are persistent scattering in which test particles are continuously interacting with a scattering center, and the Lorentz model in which a ‘light’ particle is scattered by a large number of ‘heavy’ particles The agreement between our theoretical predictions and the numerical simulations is excellent The new results are also essential in the thermodynamic limit as introduced in statistical mechanics We recover also, the results of non-equilibrium statistical mechanics obtained by various phenomenological approximations Of special interest is the domain of validity of the trajectory description as a trajectory is traditionally considered as a primitive, irreducible concept In the Liouville description the natural variables are wave vectors k which are constants in free motion and modified by interactions and resonances A trajectory can be considered as a coherent superposition of plane waves corresponding to wave vectors k Resonances correspond to non-local processes in space-time They threaten therefore the persistence of trajectories In fact, we show that whenever the thermodynamic limit exists, trajectories are destroyed and transformed into singular distribution functions We have a ‘collapse’ of trajectories, to borrow the terminology from quantum mechanics The trajectory becomes a stochastic object as in Brownian motion theory In conclusion, we obtain a unified formulation of dynamics and of thermodynamics This involves the introduction of LPS which leads to dissipation together with the consideration of delocalized situations From this point of view, there is a strong analogy with phase transitions which are also defined in the thermodynamic limit Irreversibility is, in this sense, an ‘emergent’ property which could not be included in classical dynamics as long as its study was limited to local, transient situations
71 citations
TL;DR: In this paper, the authors investigate the dynamics of a simple mechanical system and show that the structure of the bifurcation diagram seems independent of the friction models, and small changes of experimentally estimated parameters like friction and restitution coefficients (within the error of estimation) can lead to significant qualitative changes in the system's dynamics.
Abstract: Our investigations of the dynamics of a simple mechanical system show that (i) the structure of the bifurcation diagram seems to be independent of the friction models, and (ii) small changes of experimentally estimated parameters like friction and restitution coefficients (within the error of estimation) can lead to significant qualitative changes in the system's dynamics. The last observation shows that for such systems it is very difficult (if not impossible) to build mathematical models which can qualitatively describe experimental results for all possible values of system parameters.
TL;DR: In this paper, connections between the global thermodynamical interpretation of quantum mechanics and the reductionist Cantorian-fractal space-time approach are drawn between the two fields and explain how Cantorian space can serve as a geometrical model for a space time support of the thermodynamic approach to quantum mechanics.
Abstract: Connections are drawn between the global thermodynamical interpretation of quantum mechanics and the reductionist Cantorian-fractal space-time approach. The objective is to show the influence of the thermodynamical approach on development in both fields and to explain how Cantorian space can serve as a geometrical model for a space-time support of the thermodynamical approach to quantum mechanics. Seen through both theories, quantum mechanics could appear to be the result of a turbulent but homogeneous diffusion process in a transfinite non-smooth micro space-time with an area-like quantum ‘path’. Time symmetry breaking is then a consequence of the transfinite information barrier of Cantorian space-time. An important result found here is that the four dimensionality of micro space-time is a consequence of a discrete Maxwell-Boltzmann distribution of the elementary Cantor sets forming this space. In fact, it will be shown here that many of the paradoxes of quantum mechanics can be traced back to the contraintuitive character of the underlying unstable and nonsmooth Cantorian geometry of micro space-time.
TL;DR: In this article, the fractal dimension of the damaged microstructure and the exponent of a power-constitutive relation are defined as material constants, which turn out to be scale invariant material constants.
Abstract: Physicists have often observed a scaling behaviour of the main physical quantities during experiments on systems exhibiting a phase transition. The main assumption of a scaling theory is that these characteristic quantities are self-similar functions of the independent variables of the phenomenon and, therefore, such a scaling can be interpreted be means of power-laws. Since a characteristic feature of phase transitions is a catastrophic change of the macroscopic parameters of the system undergoing a continuous variation in the system state variables, the phenomenon of fracture of disordered materials can be set into the wide framework of critical phenomena. In this paper new mechanical properties are defined, with non integer physical dimensions depending on the scaling exponents of the phenomenon (i.e. the fractal dimension of the damaged microstructure, or the exponent of a power-constitutive relation), which turn out to be scale-invariant material constants. This represents the so-called renormalization procedure, already proposed in the statistical physics of random process.
TL;DR: In this paper, the motion of a slender rigid block with a flat or concave base resting on a rigid and flat foundation is analyzed and the number of impacts prior to overturning is computed, and results for horizontal foundation acceleration are plotted in the plane of excitation amplitude versus excitation frequency.
Abstract: The motion of a slender rigid block with a flat or concave base resting on a rigid and flat foundation is analyzed. The block may be symmetric or asymmetric, and the foundation may be horizontal or tilted. The foundation oscillates harmonically for a finite period of time, and the block exhibits planar motion: it may rotate about either of its bottom corners, it may rock from one corner to the other, and it may overturn. Sliding and bouncing are not considered. Energy is lost during the impact when the point of rotation switches from one corner to the other. The number of impacts prior to overturning is computed, and results for horizontal foundation acceleration are plotted in the plane of excitation amplitude versus excitation frequency. The boundaries separating regions associated with different numbers of impacts, and in particular the boundary between overturning and nonoverturning regions, are fractal.
TL;DR: In this paper, the velocity and width of ion-acoustic waves in a collisionless unmagnetized plasma were investigated both theoretically and experimentally, and the solitary wave solutions obtained by a pseudo-potential (PP) method were compared with those of a Korteweg-de Vries (KdV) equation deduced by a reductive perturbation method including the finite temperature of ions.
Abstract: The velocity and width of ion-acoustic waves in a collisionless unmagnetized plasma are investigated both theoretically and experimentally. The solitary wave solutions obtained by a pseudo-potential (PP) method are compared with those of a Korteweg-de Vries (KdV) equation deduced by a reductive perturbation method including the finite temperature of ions. For the velocity of solitons, both results are nearly equal. The square of the width times the height is constant for KdV solitons; however, it decreases with the soliton height for the PP solitary waves. The experiment has been performed by a multi-dipole double plasma machine. The velocity and width of solitary waves are measured as a function of the amplitude which is estimated taking two components of electrons into consideration. Experimental results agree well with the predictions of the PP method.
TL;DR: In this article, the largest Lyapunov exponent in daily data for the Swedish Krona vs Deutsche Mark, ECU, U.S. Dollar and Yen exchange rates is presented.
Abstract: Detecting the presence of deterministic chaos in economic time series is an important problem that may be solved by measuring the largest Lyapunov exponent. In this paper we present estimates of the largest Lyapunov exponent in daily data for the Swedish Krona vs Deutsche Mark, ECU, U.S. Dollar and Yen exchange rates. In order to estimate the dimension of the systems producing these exchange rate series, we also present estimates of the correlation dimension. We found indications of deterministic chaos in all exchange rate series. However, the estimates for the largest Lyapunov exponents are not reliable, except in the Swedish Krona-ECU case, because of the limited number of data points. In the Swedish Krona-ECU case, we found indications of a low-order chaotic dynamical system.
TL;DR: In this paper, the authors established a differential equation of motion of a nonlinear viscoelastic beam based on a novel and sophisticated stress-strain law for polymers and examined a periodically forced oscillation of such a simply supported beam and search for possible chaotic responses.
Abstract: The differential equation of motion of a nonlinear viscoelastic beam is established and is based on a novel and sophisticated stress-strain law for polymers. Applying this equation we examine a periodically forced oscillation of such a simply supported beam and search for possible chaotic responses. To this purpose we establish the Holmes-Melnikov boundary for the system. All further investigations are developed by means of a computer simulation. In this connection the authors examine critically the Poincare mapping and the Lyapunov exponent techniques and distinguish in this way between chaotic and regular motion, A set of control parameters of the equation is found, for which either a chaotic or a regular motion can be generated, depending on the initial conditions and the corresponding basins of attraction. Thus, in this particular case two attractors of completely different nature—regular and chaotic, respectively—coexist in the phase space. The basins of attraction of the two attractors for a fixed instant of time are plotted, and appear to possess a very complex fractal geometry.
TL;DR: In this article, a detailed description of bifurcation phenomena in the Henon map is presented, which strongly supports the idea that the map contains all possible bifurbcation phenomena known for two-dimensional discrete maps.
Abstract: Using the analysis of bifurcations approach the detailed description of bifurcation phenomena in the classical Henon map is presented. This description strongly supports the idea that the Henon map contains all possible bifurcation phenomena known for two-dimensional discrete maps. It is interesting to note that the existence of two different equilibria in the Henon map generates additional — dual — appearance of bifurcation phenomena. The proposed analysis can serve as a prototype of the bifurcation analysis for finite-dimensional iterative processes with multiple equilibria.
TL;DR: In this article, the Peano-Moore curve is used to visualize the qualitative behavior of particles moving on fractal trajectories in space and time, and the Schrodinger equation is also derived in terms of ensembles of classical particles and this unifies the two equations conceptually in a very direct way.
Abstract: We illustrate some of the ideas involved in fractal space-time using familiar deterministic fractals. Starting with the objective of reproducing the Heisenburg uncertainty principle for point particles, we use the Peano-Moore curve to help visualize the qualitative behaviour of particles moving on fractal trajectories in space and time. With this qualitative picture in mind we then explore exactly solvable models to verify that our ideas are mathematically consistent. We find that the Schrodinger equation describes ensembles of classical particles moving on fractal random walk trajectories. This shows that the Schrodinger equation has a straightforward microscopic model which is not, however, appropriate for quantum mechanics. The free particle Dirac equation is also derivable in terms of ensembles of classical particles and this unites the two equations conceptually in a very direct way. In both cases what we discover is a many-particle simulation of quantum mechanics and this confirms in a graphic way that the mysteries surrounding quantum mechanics lie not in the equations, but in interpretation and the theory of measurement. Finally, we discuss an exactly solvable model which incorporates fractal time. The calculation produces the Dirac equation in 1 + 1 dimensions and because of intrinsic space-time loops, constitutes a model with the potential to exhibit the wave-particle duality found in nature.
TL;DR: In this article, a system of coupled integrable dispersionless (CID) equations is considered and the integrability properties through Painleve (P) analysis are discussed.
Abstract: Considering a system of coupled integrable dispersionless (CID) equations, we discuss the integrability properties through Painleve (P) analysis. Further, we use the bilinear transformations in which nonlinear coupled dispersionless equations are modified into bilinear forms through dependent variable transformations.
TL;DR: In this paper, the authors show that chaos may emerge as a solution to a dynamic linear programming problem with infinite time-horizon problems. But they do not consider the problem of finding a solution, which cannot be derived from a simple repetition of arithmetics.
Abstract: Chaotic phenomena have been observed in various fields of sciences. We are concerned with linear programming (LP) and demonstrate that chaos may emerge as a solution to a dynamic LP problem. For this purpose, we work with an infinite time-horizon problem, for chaos appears in a dynamical system with no terminal date. As a result, it is not straightforward to find a solution, which cannot be derived from a simple repetition of arithmetics. In the finite time-horizon case, in contrast, a solution can be, at least in theory, obtained by such a method; the simplex method is one such procedure, repeating computations systematically.
TL;DR: In this article, the higher order KdV equation with a source was considered and a Backlund transformation and the nonlinear superposition formula were presented, where the source was assumed to be a source.
Abstract: The higher order KdV equation with a source is considered. A Backlund transformation and the nonlinear superposition formula are presented.
TL;DR: In this article, a deterministic method for estimating the maximum, or least favorable frequency, and the minimum, or best favorable frequency of structures with uncertain but non-random parameters is discussed.
Abstract: In this study, a new, deterministic method is discussed for estimating the maximum, or least favorable frequency, and the minimum, or best favorable frequency, of structures with uncertain but non-random parameters. The favorable bound estimate is actually a set in eigenvalue space rather than a single vector. The obtained optimum estimate is the smallest calculable set which contains the uncertain system eigenvalues. This kind of eigenvalue problem involves uncertain but non-random interval stiffness and mass matrices. If one views the deviation amplitude of the interval matrix as a perturbation around the nominal value of the interval matrix, one can solve the generalized eigenvalue problem of the uncertain but non-random interval matrices. By applying the interval extension matrix perturbation formulation, the interval perturbation approximating formula is presented for evaluating interval eigenvalues of uncertain but non-random interval stiffness and mass matrices. A perturbation method is developed which allows one to calculate eigenvalues of an uncertain but non-random interval matrix pair that always contains the system's true stiffness and mass matrices. Inextensive computational effort is a characteristic of the presented method. A numerical example illustrates the application of the proposed method.
TL;DR: In this article, the fractal properties of simple vocal sounds such as Japanese vowels were examined by evaluating fractal dimension related to the self-affine property and the existence of chaos in the attractors reconstructed from the vocal sound waveforms by evaluating the Lyapunov exponents.
Abstract: In this work, we shall examine the fractal properties of simple vocal sounds such as Japanese vowels by evaluating the fractal dimension related to the self-affine property. We shall also examine the existence of chaos in the attractors reconstructed from the vocal sound waveforms by evaluating the Lyapunov exponents. The reconstructed attractors are also examined for multifractal properties. To characterize the fractal properties of complicated vocal sounds, such as speech utterances composed of several vowels, phonemes, etc., we shall propose the time-dependent fractal dimensions (TDFDs), where the fractal dimensions are evaluated based on the self-affine property, and the time-dependent multifractal dimensions (TDMFDs). We shall then use these fractal properties in a speech recognition model to examine if our method is able to characterize complicated vocal sounds effectively. For comparison, we shall utilize the running power spectrum (RPS) as a recognition parameter. It was found that utilizing the fractal properties of vocal sounds as recognition parameters gives a high recognition rate, showing that complicated vocal sounds can be effectively characterized by their fractal properties.
TL;DR: In this paper, the bifurcation characteristics of the forced van der Pol oscillator on a specific parameter plane, including intermediate parameter regions, are investigated and the successive arrangement of the dominant mode-locking regions, where a single subharmonic solution with the rotation number, 1 (2k + 1), exists, and the transitional zones between them are depicted.
Abstract: In this paper, the bifurcation characteristics of the forced van der Pol oscillator on a specific parameter plane, including intermediate parameter regions, are investigated. The successive arrangement of the dominant mode-locking regions, where a single subharmonic solution with the rotation number, 1 (2k + 1) , exists, and the transitional zones between them are depicted. The transitional zones are explicitly proposed to be classified into two groups according to the different global characters: (1) the simple transitional zones, in which coexistence of two mode-locked solutions with rotation numbers 1 (2k ± 1) appear; (2) the complex transitional zones, in which the sub-zones with the mode-locked solutions, whose rotation numbers are rational fractions between 1 (2k + 1) and 1 (2k − 1) , and the quasi-periodic solutions exist. The emphasis of this paper is to study the evolution of the global structures in the transitional zones. A complex transitional zone generally evolves from a Farey tree, when the forcing amplitude is small, to a chaotic regime, when forcing amplitude is sufficiently large. It is of great interest that the sub-zone with a rotation number, 1 2k , which has the largest width within a complex transitional zone, can usually intrude into the dominant regions of 1 (2k − 1) before it completely vanishes. Moreover, the features of overlaps of mode-locking sub-zones and the number of coexistence of different attractors are also discussed.
TL;DR: In this paper, the fate of output in the Cournot oligopoly model when the equilibrium is locally unstable is considered, and types of nonlinearities which may be present to bound the motion and introduce time lags in production and information serve as bifurcation parameters.
Abstract: We consider the fate of output in the Cournot oligopoly model when the equilibrium is locally unstable. We discuss types of nonlinearities which may be present to bound the motion and introduce time lags in production and information which serve as bifurcation parameters. We apply the Hopf bifurcation theorem to determine conditions under which limit cycle motion is born, and use computer simulations to investigate the nature of the attractors generated by such models.
TL;DR: A novel technique for improving the performance of Lyapunov exponent calculations from a time series and results for the algorithm's performance in the presence of noise corruption are shown.
Abstract: In this paper we present a novel technique for improving the performance of Lyapunov exponent calculations from a time series. We show results for the algorithm's performance in the presence of noise corruption.
TL;DR: In this article, general forms for polynomial functions that yield attractors having the symmetry of the cube are developed and finite generating sets for these functions are also determined, applied experimentally to produce attractors that appear as points, curves, clouds, surfaces and strange attractors.
Abstract: General forms for polynomial functions that yield attractors having the symmetry of the cube are developed. Finite generating sets for these functions are also determined. These functions are applied experimentally to produce attractors that appear as points, curves, clouds, surfaces and strange attractors.
TL;DR: In this article, a modified and parametric Hurst exponent using fixed amplitudes and sampling intervals given by run length statistics is proposed, which allows us to calculate the Hurst expander for the Wiener process, its discrete time equivalent as well as a birth-death random walk.
Abstract: This paper considers a modified and parametric Hurst exponent using fixed amplitudes and sampling intervals given by run length statistics. Such an approach allows us to calculate a modified and theoretical Hurst exponent based on run length statistics. We then calculate the Hurst exponent for the Wiener process, its discrete time equivalent as well as a birth-death random walk.