Self-generated chaotic behavior in nonlinear mechanics
- ... The break-up of resonant tori and the KAM theorem, on the other hand, account for the rich dynamics in the internal toroidal region. Our computations suggest a general picture for this region, which consists of many periodic orbits of elliptic or hyperbolic type, homoclinic orbits to the hyperbolic orbits, invariant 2-tori around the elliptic ones, cantori and chaotic zones (Helleman 1980; MacKay 1994). That is, the dynamics in the interior of stationary vortex breakdown bubbles are entirely consistent with those of a mildly perturbed, volume-preserving toroidal flow and are accurately represented by Helleman's (1980) sketch of the phase-space dynamics of a perturbed Hamiltonian system. ...... Our computations suggest a general picture for this region, which consists of many periodic orbits of elliptic or hyperbolic type, homoclinic orbits to the hyperbolic orbits, invariant 2-tori around the elliptic ones, cantori and chaotic zones (Helleman 1980; MacKay 1994). That is, the dynamics in the interior of stationary vortex breakdown bubbles are entirely consistent with those of a mildly perturbed, volume-preserving toroidal flow and are accurately represented by Helleman's (1980) sketch of the phase-space dynamics of a perturbed Hamiltonian system. ...We study the motion of non-diffusive, passive particles within steady, three-dimensional vortex breakdown bubbles in a closed cylindrical container with a rotating bottom. The velocity fields are obtained by solving numerically the three-dimensional Navieril'nikov (1965). Chaotic Šil'nikov orbits spiral upward (from the spiral-in to the spiral-out saddle) around the axis and then downward near the surface, wrapping around the toroidal region in the interior of the bubble. Poincaré maps reveal that the dynamics of this region is rich and consistent with what we would generically anticipate for a mildly perturbed, volume-preserving, three-dimensional dynamical system (MacKay 1994; Mezic & Wiggins 1994a). Nested KAM-tori, cantori, and periodic islands are found embedded within stochastic regions. We calculate residence times of upstream-originating non-diffusive particles and show that when mapped to initial release locations the resulting maps exhibit fractal properties. We argue that there exists a Cantor set of initial conditions that leads to arbitrarily long residence times within the breakdown region. We also show that the emptying of the bubble does not take place in a continuous manner but rather in a sequence of discrete bursting events during which clusters of particles exit the bubble at once. A remarkable finding in this regard is that the rate at which an initial population of particles exits the breakdown region is described by the devil's staircase distribution, a fractal curve that has been already shown to describe a number of other chaotic physical systems.
- ... Another universal fractal number, which is connected with the flat geometry of a quadratic Henon map, is δ 1 = 8.7210972[14][15][16]. Numerically, δ 1 is approximately related to φ-2 by ...In fairly good agreement with the consensus range of dark energy to matter this ratio of the critical density is suggested to be connected with the golden mean φ = 0.6180339887, yielding for dark energy to matter mass fractions ΩΛ/Ωm = 72.361/27.639 = 1+φ-1 = φ-2 = 2.6180339. Assuming the baryonic matter to be only 4.432 %, the ratio of matter to baryonic matter would be Ωm/Ωb = 6.236068 = φ-3+2 = 2φ-2+1, and further the ratio of dark matter to baryonic one Ωdm/Ωb = 5.236068 = φ-3+1 = 2φ-2. If one subtracts from the dark matter a contribution of antimatter with the same mass of baryonic matter, according to the antigravity theories of Villata respectively Hajducovic, the remaining mass ratio would yield Ωr/Ωb = 4.236068 = φ-3 = 2φ-2-1. Replacing the ‘Madelung’ constant α of Villata’s ‘lattice universe’ by φ, one reaches again 1+φ as the ratio of the repulsive mass contribution to the attractive one. Assuming instead of a 3D lattice a flat 2D one of rocksalt type, the numerical similarity between the Madelung constant α2D(NaCl) = 1.6155426… and φ-1 could not be just coincidence. The proposed scaling of the cosmological mass fractions with the most irrational universal number φ may indicate that the chaotic cosmological processes have reached a quite stable equilibrium. However, the intimate connection of φ with its reciprocal may ignite the discussion whether our universe is intertwined with another universe or even part of a multiverse with the dark constituents contributed from there.
- ... The scaling works as long as the GLE holds. It is, generally, possible to derive a GLE for the Markovian systems by eliminating variables in which effects are incorporated in the memory-friction kernel and in the colored noise [50]. In particular, the absence of a coupling to a thermal bath (explicit in the GLE) and consequently, the lack of a detailed balance relation or FDT may require a specific analysis of each case. ...Diffusion regimes most frequently found in nature are described in terms of asymptotic behaviors. In this work, we use a generalization of the final- value theorem for Laplace transform in order to investigate the anomalous diffusion phenomenon for asymptotic times. We generalize the concept of the diffusion exponent, including a wide variety of asymptotic behaviors than the power law. A method is proposed to obtain the diffusion coeffi- cient analytically through the introduction of a time scaling factor, λ. We obtain as well an exact expression for λ, which makes possible to describe all diffusive regimes featuring a universal parameter determined by the dif- fusion exponent. We show the existence of two kinds of ballistic diffusion, ergodic and non-ergodic. The method is general and may be applied to many types of stochastic problem.
- ... δ = known as a universal scaling constant for two-dimensional maps in the theory of fractal systems or chaotic ones, with the precise value of δ 1 = 8.7210972@BULLET@BULLET@BULLET [16] [17]. Recently, Savin et al. [18] studied the self-oscillating system of the Van der Pol oscillator [19] subjected to an external force to compensate dissipation. ...The empirical relation of Tco (K ) = 2740 / < qc>4 between the transition temperature of optimum doped superconductors Tco and the mean cation charge c, a physical paradox, can be recast to strongly strongly support fractal theories of high- Tc superconductors, thereby applying the finding that the optimum hole concentration of σo = 0.229 can be linked with the universal fractal constant δ1 = 8.72109… of the renormalized quadratic Hénon map. The transition temperature obviously in- creases steeply with a domain structure of ever narrower size, characterized by Fibonacci num- bers. However, also conventional BCS superconductors can be scaled with δ1, exemplified through the energy gap relation kBTc ≈ 5Δ0/δ1, suggesting a revision of the entire theory of superconductiv- ity. A low mean cationic charge allows the development of a frustrated nano-sized fractal structure of possibly ferroelastic nature delivering nano-channels for very fast charge transport, in common for both high-Tc superconductor and organic-inorganic halide perovskite solar materials. With this backing superconductivity above room temperature can be conceived for synthetic sandwich structures of less than 2+. For instance, composites of tenorite and cuprite respectively tenorite and CuI (CuBr, CuCl) onto AuCu alloys are proposed. This specification is suggested by previously described filamentary superconductivity of “bulk” CuO1−x samples. In addition, cesium substitution in the Tl-1223 compound is an option.
- ... The analytic formulae based on chromatic beta-functions and higher order dispersion are rather lengthy, and codes for minimizing them run more efficient by obtaining the values from numerical differentiation of the dispersive closed orbit [3]. Amplitude dependant tune shifts lead to a twist in phase space and subsequent resonance overlap, and with it chaotic motion and fractal dynamic acceptance structure [7]. But a closed expression suitable for minimization is available (eq.(119) in [3] or eq. ...Correction of a light source's large chromaticity while mai ntaining sufficient dynamic aperture and momentum acceptance requires careful optimization of the sextupole configuration in first and second order of sextu pole strength. In this paper we try to explain chromaticity and its correcti on in a most intu- itive way. Accompanied by the step wise improvement of a test lattice, the scheme is then expanded from plain correction with two sextupole families to installation of "harmonic" setupole for first and second o rder optimization. The approach is most visual and pragmatic on expense of the mathematical formalism which may be found in the references.
- ... Beginning with the value of optimum h + of σ o = 0.229 (or multiples) [15] that is near 3/13 and is attributed to a large group of high-T c superconductors based on Tl or Hg, respectively. Surprisingly, the multiplier, which would give two holes needed to create a pair, emerges as the number known as a universal scaling constant for two-dimensional maps in the theory of fractal systems or chaotic ones, with the precise value of δ 1 = 8.7210972…[16] [17]. Recently, Savin et al. [18] studied the self-oscillating system of the Van der Pol oscillator [19] subjected to an external force to compensate dissipation. ...The empirical relation of T_co(K)=2740/ _c^4 between the transition temperature of optimum doped superconductors T_co and the mean cationic charge _c, a physical paradox, can be recast to strongly support fractal theories of high-T_c superconductors, thereby applying the finding that the optimum hole concentration of h^+ = 0.229 can be linked with the universal fractal constant delta_1 = 8.72109... of the renormalized Henon map. The transition temperature obviously increases steeply with a domain structure of ever narrower size, characterized by Fibonacci numbers. With this backing superconductivity above room temperature can be conceived for synthetic sandwich structures of _c less than 2+. For instance, composites of tenorite and cuprite respectively tenorite and CuI (CuBr, CuCl) onto AuCu alloys are proposed. This specification is suggested by previously described filamentary superconductivity of 'bulk' CuO_1-x samples. In addition, cesium substitution in the Tl-1223 compound is an option. A low mean cationic charge allows the development of a frustrated nano-sized fractal structure of possible ferroelastic nature delivering nano-channels for the very fast charge transport, in common for both high-T_c superconductors and organic inorganic halide perovskite solar materials.
- ... The dynamics of seismic waves trapped within a geological resonator of arbitrary geometry is fairly more complicated than that, and includes the possibility of the existence of irregular, chaotic or non-integrable motions (Sorauf & Rial 1988; Rial & Sorauf 1987). Furthermore, an important numerical result is that in the presence of irregular motions the nodal surfaces of the eigenfunctions do not follow an orderly, geometrical pattern as in the case of stable resonances, but appear to wander randomly across the domain (see for instance Berry 1981Berry , 1983Berry , 1987 Helleman 1980; Lichtenberg & Lieberman, 1983, ch. 6) and the eigenvalues are not described by deterministic formulae. This is the topic of a forthcoming paper. ...The resonant eigenfrequencies of three-dimensional models of sedimentary basins are calculated through the use of high-frequency asymptotics. The results are valid for the areas near the geometrical centre of the basin, where the thickness of sediments is greatest. By analogy with the theory of electromagnetic resonators it is shown that the modes of oscillation of trapped seismic waves can be easily computed from simple formulas that relate the geometry of the basin to the type of resonant mode and to the eigenfrequencies. All results are approximations based on WKB solutions to the wave equation whose phase function satisfy resonant or ‘quantum’ conditions inside the basin. Comparison with published numerical data indicates that the analytical approach here described may be of more general application than the empirical formulations that have been proposed previously for two-dimensional basins.
- ... It turns out that the period-doubling renormalization for area-preserving maps is very different from the dissipative case. A universal period-doubling cascade in families of area-preserving maps was observed by several authors in the early 80's (Derrida and Pomeau 1980, Helleman 1980, Benettin et al 1980, Bountis 1981, Collet et al 1981, Eckmann et al 1982). The existence of a hyperbolic fixed point for the period-doubling renormalization operator ...Area-preserving maps have been observed to undergo a universal period-doubling cascade, analogous to the famous Feigenbaum-Coullet-Tresser period doubling cascade in one-dimensional dynamics. A renormalization approach has been used by Eckmann, Koch and Wittwer in a computer-assisted proof of existence of a conservative renormalization fixed point. Furthermore, it has been shown by Gaidashev, Johnson and Martens that infinitely renormalizable maps in a neighborhood of this fixed point admit invariant Cantor sets with vanishing Lyapunov exponents on which dynamics for any two maps is smoothly conjugate. This rigidity is a consequence of an interplay between the decay of geometry and the convergence rate of renormalization towards the fixed point. In this paper we prove a result which is crucial for a demonstration of rigidity: that an upper bound on this convergence rate of renormalizations of infinitely renormalizable maps is sufficiently small.
- We suggest to use subharmonic stroboscopic sampling to resolve high-order bifurcations in a numerical study of driven nonlinear oscillators. A bifurcation sequence up to the 1024th period has been located in the forced Brusselator by using this method. On leave from the Institute of Physics, Academia Sinica, Beijing, China.
- We derive an equation governing the nonlinear propagation of a linearly polarized Alfvén wave in a two-dimensional, anisotropic, slightly compressible, highly magnetized, viscous plasma, where nonlinearities arise from the interaction of the Alfvén wave with fast and slow magnetoacoustic waves. The phase mixing of such a wave has been suggested as a mechanism for heating the outer solar atmosphere (Heyvaerts and Priest, 1983).We find that cubic wave damping dominates shear linear dissipation whenever the Alfvén wave velocity amplitude δvy exceeds a few times ten metres per second. In the nonlinear regime, phase-mixed waves are marginally stable, while non-phase-mixed waves of wavenumber ka are damped over a timescale kuRe 0|δ vy/vA |−2, Re 0 being the Reynolds number corresponding to the Braginskij viscosity coefficient η0 and vA the Alfvén speed. Dissipation is most effective where β = (vs /vA) 2 ≈ 1, vs being the speed of sound.
- A relevant application of the recent developments in nonlinear dynamic theory lies in providing the description and consequent calculation of the interactions among charged particles that belong either to a given beam or to different beams in collision with each other in storage rings. This paper focuses on the weak-strong version of beam-beam interactions and further limits itself to only those interactions which occur in proton-proton and proton-antiproton rings. The motion of a test particle around its main reference orbit is analyzed by conservative mapping following from periodic kicks induced by the forces of beam-beam collisions which solicit otherwise regular betatron oscillations. Also considered is the effective spin polarization of the beams after long residence in the machine in the presence of depolarizing forces. Both fixed-angle and head-on collisions are treated. The Birkhoff procedure applied in a generalized form to the mapping consists of searching for a formal simplectic transformation which reduces the mapping to normal form or to the form of integrable mapping.
- Confinement of charged particles in electromagnetic fields, plasma heating, intermolecular dynamics, etc. can all be modeled by Hamiltonian systems dq/dt=∂H(q,p,t)/∂p dp/dt=−∂H(q,p,t)/∂q where q and p are n−dimensional, H is the Hamiltonian, n the number of degrees of freedom, and (p,q) is the phase space. These lectures are structured to describe such Hamiltonian systems in the simplest non−trivial case, i.e., two degree of freedom. In this case the systems can be reduced to iteration of one− parameter families of area−preserving maps of a surface to itself: (qn + 1,pn + 1)=F(qn,pn). Periodic orbits, invariant circles, and cantori are mappings which focus this discussion. (AIP)
- We present calculations on the sticking of hyperthermal HCl to the basal plane (0001) face of ice Ih at normal and off-normal incidence. The dependence of the sticking probability on the incidence energy (Ei), the angle of incidence (θi), and the surface temperature (Ts) is discussed. Two sticking mechanisms are observed. For θi ⩽ 30°, penetration of the (0001) face is possible at an energy of about 100 kJ/mol, which is an order of magnitude lower than energies for which the penetration of metallic or covalently bonded crystals by atoms becomes possible. This possibility is due to the open structure of single-crystalline ice Ih, in which the water molecules are arranged in superimposed hexagons, forming shafts running perpendicular to the ice surface. The penetration mechanism is operative for the entire range of Ts studied (110–190 K). The second sticking mechanism, i.e., adsorption, occurs for all Ei, θi, and Ti. For θi<45°, the adsorption probability increases with θi as would be expected, because the normal component of Ei that needs to be transferred to the surface for sticking to occur scales with cos2 θi. However, for θi ≥ 45°, the adsorption probability decreases with θi. The energy transfer from HCl to the ice surface and the energy dissipation within the surface are found to be fast and efficient at normal incidence. © 2001 American Institute of Physics.
- Mathematical tools for the description of chaotic phenomena in physical systems are described and demonstrated, summarizing in part the principles presented in the author's book-length treatise on chaotic vibrations (Moon, 1987). Consideration is given to phase-plane and pseudo-phase-plane techniques, bifurcation diagrams, FFTs, autocorrelation functions, single and double Poincare maps, reduction to one-dimensional maps, Liapunov exponents, fractal dimensions, invariant distributions, chaos diagrams, and basin-boundary diagrams. The results obtained by application of these methods to data from typical mechanical and electronic oscillation experiments are presented graphically and discussed in detail.
- This article is a methodological manual for those who are interested in chaotic dynamics. An exposition is given on the foundations of the theory of deterministic chaos that originates in classical mechanics systems. Fundamental results obtained in this area are presented, such as elements of the theory of nonlinear resonance and the Kolmogorov–Arnol'd–Moser theory, the Poincaré–Birkhoff fixed-point theorem, and the Mel'nikov method. Particular attention is given to the analysis of the phenomena underlying the self-similarity and nature of chaos: splitting of separatrices and homoclinic and heteroclinic tangles. Important properties of chaotic systems — unpredictability, irreversibility, and decay of temporal correlations — are described. Models of classical statistical mechanics with chaotic properties, which have become popular in recent years — billiards with oscillating boundaries — are considered. It is shown that if a billiard has the property of well-developed chaos, then perturbations of its boundaries result in Fermi acceleration. But in nearly-integrable billiard systems, excitations of the boundaries lead to a new phenomenon in the ensemble of particles, separation of particles in accordance their velocities. If the initial velocity of the particles exceeds a certain critical value characteristic of the given billiard geometry, the particles accelerate; otherwise, they decelerate.
- We study the chaotic mixing in two periodic model flows, the ‘tendril–whorl’ flow and the ‘Aref-blinking-vortex’ flow, with the objective of supplying evidence for the primary mechanisms responsible for mixing in two-dimensional deterministic flows. The analysis is based on tools of dynamical systems theory but it is clear that the mixing problem generates several questions of its own: low periodic points and horseshoes dominate the picture, since we want to achieve mixing quickly; Poincaré sections, popular in dynamical systems analyses, might give misleading information with regard to dispersion at short times. Our analysis shows that both flows are able to stretch and fold material lines well below the lengthscale of the flows themselves. The inner workings of the two systems are revealed by studying the local and global bifurcations. Computations for the blinking-vortex system indicate the existence of an optimum period at which the average efficiency is maximized, whereas the intensity of segregation – a classical parameter in mixing studies – decays rapidly to an asymptotic value in the globally chaotic region. Even though our flows are not turbulent the results might have some implications for pointing to the limits of similar studies in actual turbulent flows (e.g. line stretching).
- Even though the first theoretical example of chaotic advection was a three-dimensional flow (Hénon 1966), the number of theoretical studies addressing chaos and mixing in three-dimensional flows is small. One problem is that an experimentally tractable three-dimensional system that allows detailed experimental and computational investigation had not been available. A prototypical, bounded, three-dimensional, moderate-Reynolds-number flow is presented; this system lends itself to detailed experimental observation and allows high-precision computational inspection of geometrical and dynamical effects. The flow structure, captured by means of cuts with a laser sheet (experimental Poincaré section), is visualized via continuously injected fluorescent dye streams, and reveals detailed chaotic structures and chains of high-period islands. Numerical experiments are performed and compared with particle image velocimetry (PIV) and flow visualization results. Predictions of existing theories for chaotic advection in three-dimensional volume-preserving flows are tested. The ratio of two frequencies of particle motion – the frequency of motion around the vertical axis and the frequency of recirculation in the plane containing the axis – is identified as the crucial parameter. Using this parameter, the number of islands in the chain can be predicted. The same parameter – using as a base-case the integrable motion – allows the identification of operating conditions where small perturbations lead to nearly complete mixing.
- The structure of solutions of the Burgers equation in the inviscid case is investigated numerically by computing the space-time behavior of the asymptotic solutions expressed as sequences of triangular shock waves. They are sensitively dependent on initial conditions and can display intrinsic randomness, depending on the number of zeros of the initial velocity fields.
- A systematic way for deriving the parameter renormalization equation for one-dimensional maps is presented and the critical behavior of periodic-doubling is investigated. The estimates of accumulation points and universal constants match the known values asymptotically when the order of potential grows large. The potential function derived from the parameter renormalization equation shows scaling in the parameter space with the universal convergent rate at the accumulation point, but with a different size scaling factor from the Feigenbaum constant.
- A method is given for quantizing volume-preserving polynomial mappings, using Heisenberg's matrix formulation of quantum mechanics. The energy levels of the linear map are obtained exactly and those of the quadratic, nonintegrable, Hénon map approximately and numerically.
- The following is a set of outlines and bibliographies for lectures presented at a Summer Workshop on Nonequilibrium Phenomena held from June 22 to July 3, 1981 at the Institute for Theoretical Physics in Santa Barbara. These outlines were distributed to the participants in lieu of formal proceedings, and they are being presented for publication in the same form, in the belief that the information they contain will be useful to a wider audience. It should be clearly stated, however, that the compilation is an informal one which does not claim to be a complete survey of the subject.
- Weakly damped parametrically excited cross waves with surface tension were analyzed in order to demonstrate that cross waves are chaotic. The Wiggins-Holmes extension of the Generalized Melnikov Method (GMM) and the Generalized Herglotz Algorithm (GHA) to non-autonomous systems were applied to weakly damped parametrically excited cross waves. The non-dissipative perturbed Hamiltonian system with surface tension, which satisfies the Kolmogorov-Arnold-Moser (KAM) non-degeneracy requirements, and the Melnikov integral was calculated to demonstrate that the motion was chaotic. The chaotic motion for the perturbed dissipative system with surface tension was demonstrated by numerical computation of positive Liapunov characteristic exponents.
- For two-dimensional diffeomorphisms or flows reducing essentially to them the evolution of S.A. can be described geometrically using bifurcations, homoclinic and heteroclinic points. However, many questions are left open:1) Prediction of values of the parameters for which a S.A. appears or is suddenly destroyed. 2) Existence of invariant measures on the S.A. Ergodic or mixing properties of the diffeomorphism restricted to the S.A., with respect to this measure. 3) Examination of the geometry of the S.A. for higher dimensions. Mechanisms producing or destroying S.A. in this case: Study of homo/heteroclinic points of normally hyperbolic invariant or periodic objects. We strongly recommend to look for the geometric structure in physical or numerical experiments. It seems to us that without this knowledge one cannot get a really deep insight in the problem of S.A.
- We conduct detailed simulations in order to investigate the structure of steady vortex breakdown bubbles. Careful examination of all experimental visualizations reveals a number of ubiquitous features, such as the existence of dye inside the bubble and asymmetries in its downstream end. We show numerically that such features can be attributed to imperfections in the setup of the experimental apparati, imperfections that may be minimized with careful manufacturing and setting up of the rig, but are never completely eliminated. We study such forced breaking of symmetry for imperfections of an order of magnitude typical to those of realistic experimental setups.
- It is argued that E-infinity theory may represent the universality class of all universality classes of certain discrete dynamical maps which are at the root of relevant field theories. First we give a concise derivation of the basic equations of E-infinity and its ground state. Subsequently it is shown that the independence of the results obtained from the details of any equations of motion or Lagrangian is a clear indication that E-infinity may represent the universality class of all universality classes in the sense of Cantor with regard to relevant quantum field theories.
- A microcanonical formulation of euclidean quantum field theories is presented. In the formulation, correlation functions are given by a microcanonical ensemble average of fields. Furthermore, the perturbative equivalence of the formulation and the standard functional formulation is proved and the equipartition law is derived in our formulation.
- Deterministic mathematical models of neural systems can give rise to complex aperiodic ('chaotic') dynamics in the absence of stochastic fluctuations ('noise') in the variables or parameters of the model or in the inputs to the system. The authors show that chaotic dynamics are expected in nonlinear feedback systems possessing time delays such as are found in recurrent inhibition and from the periodic forcing of neural oscillators. The implications of the possible occurrence of chaotic dynamics for experimental work and mathematical modeling of normal and abnormal functions neurophysiology are mentioned.
- In this chapter, the basic concepts of nonlinear discrete systems will be presented. The local and global theory of stability and bifurcation for nonlinear discrete systems will be discussed. The stability switching and bifurcation on specific eigenvectors of the linearized system at fixed points under specific period will be presented. The higher singularity and stability for nonlinear discrete systems on the specific eigenvectors will be developed. A few special cases in the lower dimensional maps will be presented for a better understanding of the generalized theory. The route to chaos will be discussed briefly, and the intermittency phenomena relative to specific bifurcations will be presented. The normalization group theory for 2-D discrete systems will be presented via Duffing discrete systems.
- We now focus our attention on 1D tilings. While it is true that tilings in 1D are not as rich in structure as those in higher dimensions, the results to be obtained serve as valuable pointers to the possibilities in more involved situations.
- By Liouville’s Theorem, the solution flow for a conservative Hamiltonian system preserves volumes in phase space. Poincaré’s Recurrence Theorem then shows that most solution curves in phase space come back infinitely often arbitrarily close to the initial points. This implies that conservative systems can not asymptotically collapse onto any lower-dimensional attractors. However, conservative systems can exhibit a rich variety of structures with chaotic and regular orbits interspersed among each other, (Heileman, 1980). Chaotic motion appears via homoclinic intersections of unstable manifolds.
- In section 2.4 we were introduced to Hamiltonian systems. If H is a C2 function of the 2n variables pi , qi , i = 1, … n, H : ℝ2n → ℝ, then the equations of Hamilton(15-x) $${\dot p}_i = -{\partial H\over \partial q_i}, {\dot q}_i = {\partial H\over \partial p_i}, i = 1, “ots , n.$$Now we have for the orbital derivative Lt H = 0, so H(p, g) is a first integral of the equations 15.1. We have seen a number of examples where n = 1; in this case the integral H(p, q) = constant describes the orbits in the phase-plane completely.
- Dimension is perhaps the most basic property of an attractor. In this paper we discuss a variety of different definitions of dimension, compute their values for a typical example, and review previous work on the dimension of chaotic attractors. The relevant definitions of dimension are of two general types, those that depend only on metric properties, and those that depend on the frequency with which a typical trajectory visits different regions of the attractor. Both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value, which we call the “dimension of the natural measure”, and all of the metric dimensions take on a common value, which we call the “fractal dimension”. Furthermore, the dimension of the natural measure is typically equal to the Lyapunov dimension, which is defined in terms of Lyapunov numbers, and thus is usually far easier to calculate than any other definition. Because it is computable and more physically relevant, we feel that the dimension of the natural measure is more important than the fractal dimension.
- A new edition of this well-established monograph, this volume provides a comprehensive overview over the still fascinating field of chaos research. The authors include recent developments such as systems with restricted degrees of freedom but put also a strong emphasis on the mathematical foundations. Partly illustrated in color, this fourth edition features new sections from applied nonlinear science, like control of chaos, synchronisation of nonlinear systems, and turbulence, as well as recent theoretical concepts like strange nonchaotic attractors, on-off intermittency and spatio-temporal chaotic motion. © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. All rights reserved.
- An exciting new development in nonlinear dynamics, period-doubling to chaos[58,59,1], is derived here, approximately but analytically, for conservative as well as dissipative systems.
- It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of R2. A renormalization approach has been used in (Eckmann et al 1982) and (Eckmann et al 1984) in a computer-assisted proof of existence of a “universal ” areapreserving map F ∗ — a map with orbits of all binary periods 2k, k ∈ N. In this paper, we consider infinitely renormalizable maps — maps on the renormalization stable manifold in some neighborhood of F ∗ — and study their dynamics. For all such infinitely renormalizable maps in a neighborhood of the fixed point F ∗ we prove the existence of a “stable ” invariant set C ∞ F such that the maximal Lyapunov exponent of F |C ∞ is zero, and whose Hausdorff dimension F satisfies dimH(C ∞ F) ≤ 0.836. We also show that there exists a submanifold, Wω, of finite codimension in the renormalization local stable manifold, such that for all F ∈ Wω the set C ∞ F is “weakly rigid”: the dynamics of any two maps in this submanifold, restricted to the stable set C ∞ F, is conjugated by a bi-Lipschitz transformation that preserves the Hausdorff dimension. Contents
- A P–D (Proportional plus Derivative) feedback controller is proposed for the control of chaotic dynamic systems. Using this feedback law the design requirement of command matching is satisfied independently of the perturbations of the initial conditions of the dynamic plant. The problem is solved for all nonlinear systems having independent inputs and independent performance variables. The controller is implemented without knowledge of the system’s initial conditions and thus it is independent of their perturbations.
- We have investigated numerically the behaviour, as a perturbation parameter is varied, of periodic orbits of some reversible area-preserving maps of the plane. Typically, an initially stable periodic orbit loses its stability at some parameter value and gives birth to a stable orbit of twice the period. An infinite sequence of such bifurcations is accomplished in a finite parameter range. This period-doubling sequence has a universal limiting behaviour: the intervals in parameter between successive bifurcations tend to a geometric progression with a ratio of 1/δ = 1/8.721097200..., and when examined in the proper coordinates, the pattern of periodic points reproduces itself, asymptotically, from one bifurcation to the next when the scale is expanded by α = -4.018076704... in one direction, and by β = 16.363896879... in another. Indeed, the whole map, including its dependence on the parameter, reproduces itself on squaring and rescaling by the three factors α, β and δ above. In the limit we obtain a universal one-parameter, area-preserving map of the plane. The period-doubling sequence is found to be connected with the destruction of closed invariant curves, leading to irregular motion almost everywhere in a neighbourhood.
- The renormalization procedure facilitates the detection of periodic and chaotic patterns of many two-dimensional systems of nonlinear equations. We first give a comprehensive presentation of this procedure. We then generalize it to make it more applicable to economic frameworks.
- Results originating in the theory of continuum kinematics are applied to the study of Lyapunov exponents. Algorithms are derived for the computation of the smallest and largest bounds and the sum of the Lyapunov exponents. These are based on kinematical theorems for the material derivative of stretch, volume, material unit vectors, hypersurface normals and area elements. The close connection between the theory of Lyapunov exponents and certain aspects of continuum kinematics is demonstrated. Numerical studies indicate the utility of the derived algorithms.
- The two-dimensional dynamics of plasma motion in an idealized form of the corotating convection system of the jovian magnetosphere is investigated. Plasma pressure is ignored, and a dipole magnetic field is assumed, with the magnetic axis and spin axis aligned. The effects of microdiffusion are incorporated into an otherwise macroscopic magnetohydrodynamic model. By applying the mathematical technique (of Galerkin truncation) that was used to derive the famous Lorenz equations for convection in Earth's atmosphere, we reduce the governing partial differential equations to simple, low-order, ordinary differential form. On the basis of this extreme simplification, the plasma dynamics of the jovian magnetosphere is found to be controlled by the height-integrated Pedersen ionospheric conductivity Σ; viz., if Σ is greater than a critical value Σcthen plasma motion is steady-state convection, whereas if Σ is less than Σcthen plasma motion is expected to be chaotic. The value Σcis found to lie in the approximate range 0.7 mho ≤ Σc≤ 1.8 mho, which is within the accepted range of values of Σ for Jupiter. This predicted form of control of magnetospheric plasma motion by the height-integrated Pedersen ionospheric conductivity is testable if sufficiently refined data become available. Lyapunov (fractal) dimensions of typical strange attractors are found to be in the rangeDL∼ 2.03–2.3. Whether a low-dimensional strange attractor does exist at certain times in the jovian magnetosphere can be reliably tested only by analyzing a sufficiently long time series constructed from plasma velocity data.
- Several examples of the one-dimensional mapping which are exactly solvable and show chaotic behaviour are presented. The importance of the accuracy of the numerical calculation is stressed.
- Results originating in the theory of continuum kinematics are applied to the study of Lyapunov exponents. Algorithms are derived for the computation of the smallest, largest, bounds and sum of the Lyapunov exponents. These are based on kinematical theorems for the material derivative of stretch, volume, material unit vectors, hypersurface normals and area elements. The close connection between the theory of Lyapunov exponents and certain aspects of continuum kinematics is demonstrated. Numerical studies indicate the utility of the derived algorithms.
- Nonlinear dynamical systems, being a realistic representation of nature, often exhibit a somewhat complicated behaviour. Their analysis requires a thorough investigation into the solutions of the governing nonlinear differential equations. In this paper, an approximate method is presented for solving complex nonlinear differential equations of the form: z̈+ω2z+εf(z,z̄,ż,z̄̇)=0,where z is a complex function and ε is a small parameter. It is based on the generalized averaging method which we have developed recently. Our approach can be viewed as a generalization of the approximate method based on the Krylov–Bogoliubov averaging method. The study of these systems is of interest to several fields of statistical mechanics, physics, electronics and engineering. Application of this method to special cases is performed for the purpose of comparison with numerical computations. Excellent agreement is found for reasonably large values of ε, which shows the applicability of this method to this kind of nonlinear dynamical systems. This agreement gives extra confidence that the analytical results are correct. These analytical results can be used as a theoretical guidance for doing further numerical or theoretical studies.
- A mechanism for period doubling and transition to chaos for the dissipative Henon map is investigated. The renormalization group technique is used for that purpose. In the context of this technique, a special approach is developed that relates the renormalization procedure with the simpler problem of the renormalization of the conservative Henon map. Keywordsrenormalization group-dissipative Henon map-transition to chaos
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