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  • 1
    Electronic Resource
    Electronic Resource
    Symplectic Tangent map for Planetary Motions (1999)
    Springer
    Celestial mechanics and dynamical astronomy 74 (1999), S. 59-67 
    Details
    ISSN: 1572-9478
    Keywords: planetary systems ; Liapunov exponents ; chaos ; symplectic integration
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract Equations are presented for the computation of tangent maps for use in nearly Keplerian motion, approximated by use of a symplectic leapfrog map. The resulting algorithms constitute more accurate and efficient methods to obtain the Liapunov exponents and the state transition matrix, and can be used to study chaos in planetary motions, as well as in orbit determination procedures from observations. Applications include planetary systems, satellite motions and hierarchical, nearly Keplerian systems in general.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008312912468
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  • 2
    Electronic Resource
    Electronic Resource
    Chaos and secondary resonances in the mimas–tethys system (1999)
    Springer
    Celestial mechanics and dynamical astronomy 74 (1999), S. 111-146 
    Details
    ISSN: 1572-9478
    Keywords: Mimas–Tethys system ; chaos ; secondary resonances ; tides ; capture probability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We have investigated the role of the 200 yr period discovered by Vienne and Duriez (1992) on the tidal evolution of the Mimas–Tethys system through the 2:4 ii′ present resonance. Three terms are found to generate this period. We present a perturbed‐pendulum model in which these terms bring about a perturbation to the ideal ii′ resonance pendulum, which is in a direct ratio to the eccentricity e′ of Tethys. Although e′ is now very small, it is shown that this quantity could have been much greater in the past. We also show, thanks to this model, that these terms may have brought about a stochastic layer of noticeable width at the time of capture in the ii′ resonance, with the consequence that the possible values of the inclination i of Mimas before capture range from 0.4° to 0.6° (these uncertainties arise from the present uncertainties on e′). The role of each one of the three terms is examined in the appearance of chaos. A capture into the 1/1 secondary resonance (between the libration period of the primary ii′ resonance and the period of about 200 yr) is found possible. It means that the system could have experienced several captures in the primary resonance, instead of a single one, and that i could have been, with this assumption, much lower than 0.4°. A probability of capture into this secondary resonance as a function of the eccentricity of Tethys on encounter is derived, using Malhotra's method (Malhotra, 1990). Allan's values of i = 0.42° and e′ ≈ 0 (Allan, 1969) are therefore called into question, and taking e′ ≠ 0 is shown to be absolutely necessary if we want to understand the phenomena at work in the Mimas–Tethys system.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008314007365
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  • 3
    Electronic Resource
    Electronic Resource
    Chaotic transitions in resonant asteroidal dynamics (1996)
    Springer
    Celestial mechanics and dynamical astronomy 64 (1996), S. 93-105 
    Details
    ISSN: 1572-9478
    Keywords: asteroids ; resonance ; chaos ; Kirkwood gaps
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The utilization of chaotic dynamics approaches allowed the identification of many modes of motion in resonant asteroidal dynamics. As these dynamical systems are not integrable, the motion modes are not separated and one orbit may transit from one mode to another. In some cases, as in the \31 resonance, these transitions may lead, in a relatively short time scale, to eccentricities so high that the asteroid may approach the Sun and be destroyed. In the \21 and \32 resonances these transitions are much slower and only indirect estimations of the time which is needed for a generic asteroid to leave the resonance are possible. It may reach hundreds of million years in the more robust regions of the \21 resonance and a time of the order of billions of years in those of the \32 resonance. These values are consistent with the observed depletion of the \21 resonance (only a few asteroids known while almost 60 asteroids are known in the \32 resonance).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00051608
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  • 4
    Electronic Resource
    Electronic Resource
    The two-body problem with radiation pressure in a rotating reference frame (1995)
    Springer
    Celestial mechanics and dynamical astronomy 61 (1995), S. 287-313 
    Details
    ISSN: 1572-9478
    Keywords: Two-body problem ; radiation pressure ; horseshoes ; chaos ; three-body problem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We study a perturbed Newtonian two-body problem, in which the perturbation is due to a force field of constant magnitude but rotating direction. By considering this system as a perturbation of the non-rotating case a Melnikov-type analysis allows us to show the existence of horseshoes in the level sets of the Hamiltonian and the subsequent sensitive dependence on initial conditions and non-integrability. We discuss the consequences of these results for a particular planar restricted three-body problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00051898
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  • 5
    Electronic Resource
    Electronic Resource
    Chaotic dynamics of planet‐encountering bodies (1998)
    Springer
    Celestial mechanics and dynamical astronomy 70 (1998), S. 181-200 
    Details
    ISSN: 1572-9478
    Keywords: chaos ; dynamics ; comets ; NEAs ; Lyapunov exponent
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The dynamics of two families of minor inner solar system bodies that suffer frequent close encounters with the planets is analyzed. These families are: Jupiter family comets (JF comets) and Near Earth Asteroids (NEAs). The motion of these objects has been considered to be chaotic in a short time scale,and the close encounters are supposed to be the cause of the fast chaos. For a better understanding of the chaotic behavior we have computed Lyapunov Characteristic Exponents (LCEs) for all the observed members of both populations. LCEs are a quantitative measure of the exponential divergence of initially close orbits. We have observed that most members of the two families show a concentration of Lyapunov times (inverse of LCE) around 50–100yr. The concentration is more pronounced for JF comets than for NEAs, among which a lesser spread is observed for those that actually cross the Earth's orbit (mean perihelion distance q 〈 1.05 AU). It is also observed that a general correspondence exists between Lyapunov times and the time between consecutive encounters. A simple model is introduced to describe the basic characteristics of the dynamical evolution. This model considers an impulsive approach, where the particles evolve unperturbedly between encounters and suffer ‘kicks’ in semimajor axis at the encounters. It also reproduces successfully the short Lyapunov times observed in the numerical integrations and is able to estimate the dynamical lifetimes of comets during a stay in the Jupiter family in correspondence with previous estimates. It has been demonstrated with the model that the encounters with the largest effect on the exponential growth of the distance between initially nearby orbits are neither the infrequent deep encounters, nor the frequent and far ones; instead, the intermediate approaches have the most relevant contribution to the error growth. Such encounters are at a distance a few times the radius of the Hill's sphere of the planet (e.g. 3). An even simpler model allows us to get analytical estimates of the Lyapunov times in good agreement with the values coming from the model above and the numerical integrations. The predictability of the medium‐term evolution and the hazard posed to the Earth by those objects are analysed in the Discussion section.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008331422678
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  • 6
    Electronic Resource
    Electronic Resource
    Experimental Investigation of Stability Enhancement in Semiconductor Lasers with Optical Feedback (1998)
    Springer
    Optical review 5 (1998), S. 280-284 
    Details
    ISSN: 1349-9432
    Keywords: semiconductor laser ; optical feedback ; chaos ; relaxation oscillation ; linear stability analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The stability enhancement of laser output power for the change of external cavity position in semiconductor lasers with optical feedback is observed by experiment. The relaxation oscillation frequency which plays an important role in the dynamics of the nonlinear system is also investigated as a function of the external cavity length. The period of the stability enhancement along the position of the external cavity is exactly coincident with the length corresponding to the relaxation oscillation frequency of the solitary laser. The experimental results are compared with theoretical and excellent coincidence between the two is found.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s10043-998-0280-9
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  • 7
    Electronic Resource
    Electronic Resource
    Dynamics of Semiconductor Lasers with Optical Feedback from Photorefractive Phase Conjugate Mirror (1999)
    Springer
    Optical review 6 (1999), S. 359-364 
    Details
    ISSN: 1349-9432
    Keywords: semiconductor laser ; photorefractive optical feedback ; instability ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The dynamics of semiconductor lasers with photorefractive phase conjugate optical feedback are experimentally studied. Photorefractive fringes considered here are rather static compared with time fluctuations of laser output power. Therefore, it is expected that a semiconductor laser with photorefractive feedback shows similar dynamics to those with conventional optical feedback. We examine relaxation oscillation and external cavity modes of laser output power in the presence of photorefractive phase conjugate feedback. It is proved that the dynamics of photorefractive phase conjugate feedback are fundamentally the same as those of conventional optical feedback.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s10043-999-0359-y
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  • 8
    Electronic Resource
    Electronic Resource
    Low-Frequency Fluctuation and Frequency-Locking in Semiconductor Lasers with Long External Cavity Feedback (1999)
    Springer
    Optical review 6 (1999), S. 399-401 
    Details
    ISSN: 1349-9432
    Keywords: semiconductor laser ; optical feedback ; low frequency fluctuation ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The properties of low-frequency fluctuations in semiconductor lasers with optical feedback from a long external cavity are experimentally studied. Frequency-locking of the laser light output to the injection current modulation is observed when the modulation frequency approaches the external cavity mode. The modulation frequency for the successful frequency-locking is always less than the external cavity mode frequency and the locking domains as a function of the modulation amplitude is asymmetric with respect to the frequency detuning.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s10043-999-0399-3
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  • 9
    Electronic Resource
    Electronic Resource
    Intermittent behaviour of volcanic tremor at Mt. Etna (1996)
    Springer
    Pure and applied geophysics 147 (1996), S. 729-744 
    Details
    ISSN: 1420-9136
    Keywords: Volcanic tremor ; intermittency ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Physics
    Notes: Abstract Harmonic tremor is widely studied and modelled in a very narrow frequency band (1–5 Hz) which represents the eigenfrequencies of a resonator assumed as the source of the phenomenon. Minimal effort was dedicated towards understanding its behaviour in larger temporal scales. Here we characterise the dynamic behaviour of volcanic tremor while evaluating the complete spectrum of the generalised dimension of the phase space. The starting time series constitutes the tremor amplitude picked every 10 minutes. The choice of this lag time is made on the basis of a qualitative analysis of the properties of the tremor. The results show intermittent behaviour of the dynamics which requires an 8-dimensional map to be completely described. An interesting result is that the maximum clustering of point density in phase space occurs in a monodimensional space which implies a periodicity sometimes observed experimentally. An appropriate predictive model needs more constraints on the nature of the eight variables involved in the process.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01089699
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  • 10
    Electronic Resource
    Electronic Resource
    Anomalous biennial oscillations in a Fisher equation with a discretized verhulst term (1995)
    Springer
    Journal of statistical physics 81 (1995), S. 761-775 
    Details
    ISSN: 1572-9613
    Keywords: Logistic map ; diffusion ; Fisher equation ; chaos ; oscillations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The dynamics of a biological population governed by a modified Fisher equation is studied by means of Monte Carlo simulations. Reproduction of the population occurs at discrete times, while transport caused by diffusion and conduction takes place on shorter time scales. The discrete reproduction, modeled with a set of coupled logistic maps, exhibits phenomena which are not evident in the usual continuum version of the Fisher equation. Several mechanisms for biennial oscillations of the total population are investigated. One of these shows an ordered coupling between random diffusive motion and the chaotic attractor of the logistic map.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02179256
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  • 11
    Electronic Resource
    Electronic Resource
    Kolmogorov–Sinai Entropy, Lyapunov Exponents, and Mean Free Time in Billiard Systems (1997)
    Springer
    Journal of statistical physics 88 (1997), S. 807-824 
    Details
    ISSN: 1572-9613
    Keywords: Billiards ; Kolmogorov–Sinai entropy ; Lyapunov exponents ; ergodic theory ; chaos ; numerical experiments
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We perform new experiments on the Kolmogorov–Sinai entropy, Lyapunov exponents, and the mean free time in billiards. We study their dependence on the geometry of the scatterers made up of two interpenetrating square lattices, each one with circular scatterers with different radius. We find, in particular, that the above quantities are continuous functions of the ratio of the scatterer radius. However, it seems that their derivative is discontinuous around the radius ratio which separates the diffusive and nondiffusive types of geometries.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/B:JOSS.0000015173.74708.2a
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  • 12
    Electronic Resource
    Electronic Resource
    Chaotic Behavior in Shell Models and Shell Maps (1998)
    Springer
    Journal of statistical physics 93 (1998), S. 833-842 
    Details
    ISSN: 1572-9613
    Keywords: Shell models ; turbulence ; chaos ; Lyapunov exponent ; conserved quantities ; shell maps
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We study the chaotic behavior of the GOY shell model by measuring the variation of the maximal Lyapunov exponent with the parameter ε which determines the nature of the second invariant (the generalized “helicity” invariant). After a Hopf bifurcation, we observe a critical point at ε c ∼0.38704 above which the maximal Lyapunov exponent grows nearly linearly. For high values of ε the evolution becomes regular again, which can be explained by a simple analytic argument. A model with few shells shows two transitions. To simplify the model substantially we introduce a shell map which exhibits similar properties as the GOY model.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/B:JOSS.0000033165.51531.8c
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  • 13
    Electronic Resource
    Electronic Resource
    Non-Self-Similar Behavior in the LSW Theory of Ostwald Ripening (1999)
    Springer
    Journal of statistical physics 95 (1999), S. 867-902 
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    ISSN: 1572-9613
    Keywords: kinetics of phase transitions ; domain coarsening ; asymptotic behavior ; self-similarity ; stability ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The classical Lifshitz–Slyozov–Wagner theory of domain coarsening predicts asymptotically self-similar behavior for the size distribution of a dilute system of particles that evolve by diffusional mass transfer with a common mean field. Here we consider the long-time behavior of measure-valued solutions for systems in which particle size is uniformly bounded, i.e., for initial measures of compact support. We prove that the long-time behavior of the size distribution depends sensitively on the initial distribution of the largest particles in the system. Convergence to the classically predicted smooth similarity solution is impossible if the initial distribution function is comparable to any finite power of distance to the end of the support. We give a necessary criterion for convergence to other self-similar solutions, and conditional stability theorems for some such solutions. For a dense set of initial data, convergence to any self-similar solution is impossible.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1004546215920
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  • 14
    Electronic Resource
    Electronic Resource
    Ergodicity, ensembles, irreversibility in Boltzmann and beyond (1995)
    Springer
    Journal of statistical physics 78 (1995), S. 1571-1589 
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    ISSN: 1572-9613
    Keywords: Boltzmann ; ergodicity ; irreversibility ; Ruelle principle ; SRB measures ; chaos ; nonequilibrium
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The contents of a not too well-known paper by Boltzmann are critically examined. The etymology of the word ergodic and its implications are discussed. A connection with the modern theory of Ruelle is attempted.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02180143
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  • 15
    Electronic Resource
    Electronic Resource
    A remark on papers by pixton and oliveira: Genericity of symplectic diffeomorphisms ofS 2 with positive topological entropy (1995)
    Springer
    Journal of statistical physics 80 (1995), S. 481-485 
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    ISSN: 1572-9613
    Keywords: Genericity ; chaos ; topological entropy
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We prove the existence of an open and dense subset of mapsfεDiff ω ∞ (S2) which have positive topological entropy. It follows that these maps have infinitely many hyperbolic periodic points and an exponential growth rate of hyperbolic periodic points. The proof is an application of Pixton's theorem
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02178369
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  • 16
    Electronic Resource
    Electronic Resource
    Chaotic properties of multipoint correlation functions of an Ising model with long-range interactions on the Sierpiński—Gasket lattice (1996)
    Springer
    Journal of statistical physics 82 (1996), S. 1099-1112 
    Details
    ISSN: 1572-9613
    Keywords: Ising model with long-range interactions ; Sierpiński-gasket lattice ; correlation functions ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract A class of multispin correlation functions of an Ising model with ferromagnetic nearest neighbor interactionsK and constant (distance-independent) long-range interactionsQ 1=Q,l=1,2,..., on the Sierpiński-gasket lattice is considered. Using an exact method for calculating thermodynamic functions of hierarchically constructed Ising systems, it is shown that, for a set of values ofQ and for almost all values ofK, someM k-spin correlation functions, whereM k=3 k +3 withk=1,2,...,n andn=1,2,... being the order of lattice construction, change chaotically asn, k, and therebyM k increase to infinity. Accordingly, in the thermodynamic limit, these correlation functions prove to be nonanalytic for appropriate values ofQ andK. SinceM k-point correlation functions withk being finite, i.e., correlation functions involving finite numbers of spins, remain analytic asn tends to infinity, there is a smooth crossover between analytic properties of correlation functions of the two types.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02179804
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  • 17
    Electronic Resource
    Electronic Resource
    Ergodic theorems arising in correlation dimension estimation (1996)
    Springer
    Journal of statistical physics 85 (1996), S. 25-40 
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    ISSN: 1572-9613
    Keywords: Glivenko-Cantelli theorem ; fractal ; almost sure convergence ; moment estimators ; least square estimators ; dynamical systems ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The Grassberger-Procaccia (GP) empirical spatial correlation integral, which plays an important role in dimension estimation, is the proportion of pairs of points in a segment of an orbit of lengthn, of a dynamical system defined on a metric space, which are no more than a distancer apart. It is used as an estimator of the GP spatial correlation integral, which is the probability that two points sampled independently from an invariant measure of the system are no more than a distancer apart. It has recently been proven, for the case of an ergodic dynamical system defined on a separable metric spaceythat the GP empirical correlation integral converges a.s. to the GP correlation integral at continuity points of the latter asn→∞. It is shown here that for ergodic systems defined on ℜd with the “max” metric the convergence is uniform inr. Further, a simplified proof based on weak convergence arguments of the result in separable spaces is given. Finally, the Glivenko-Cantelli theorem is used to obtain ergodic theorems for both the moment estimators and least square estimators of correlation dimension.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02175554
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  • 18
    Electronic Resource
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    Thermodynamic formalism and localization in Lorentz gases and hopping models (1997)
    Springer
    Journal of statistical physics 87 (1997), S. 1253-1271 
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    ISSN: 1572-9613
    Keywords: Lorentz lattice gases ; chaos ; thermodynamic formalism ; random walks ; localization transition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The thermodynamic formalism expresses chaotic properties of dynamical systems in terms of the Ruelle pressure ψ(β). The inverse-temperature-like variable β allows one to scan the structure of the probability distributin in the dynamic phase space. This formalism is applied here to a lorentz lattice gas. where a particle moving on a lattice of sizeL d collides with fixed scatterers placed at random locations. Here we give rigorous arguments that the Ruelle pressure in the limit of infinite systems has two branches joining with a slope discontinuity at β=1. The low- and high-β branches correspond to localization of trajectories on respectively the “most chaotic” (highest density) region and the “most deterministic” (lowest density) region, i.e. ψ(β) is completely controlled by rare fluctuations in the distribution of scatterers on the lattice. and it dose not carry and information on the global structure of the static disorder. As β approaches unity from either side, a localization-delocalization transition leads to a state where trajectories are extended and carry information on transprot properties. At finiteL the narrow region around β=1 where the trajectories are extended scales as (InL)−2. where α depends on the sign of 1−β, ifd〉1, and as (L InL)−1 ifd=1. This result appears to be general for diffusive systems with static disorder, such as random walks in random environments or for the continuous Lorentz gas. Other models of random walks on disordered lattices, showing the same phenomenon, are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02181283
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  • 19
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    Finite Range of Large Perturbations in Hamiltonian Dynamics (1998)
    Springer
    Journal of statistical physics 92 (1998), S. 909-972 
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    ISSN: 1572-9613
    Keywords: Perturbation theory ; Hamiltonian dynamics ; wave–particle interaction: transport properties ; chaos ; plasma turbulence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The dynamics defined by the Hamiltonian $$H = p^2 /2 + A\sum\nolimits_{m = - M}^M {\cos (q - mt + \varphi m)}$$ , where the φ m are fixed random phases, is investigated for large values of A, and for $$M \gg A^{2/3}$$ . For a given P * and for $$\Delta \upsilon \geqslant A^{2/3}$$ , this Hamiltonian is transformed through a rigorous perturbative treatment into a Hamiltonian where the sum of all the nonresonant terms, having a Q dependence of the kind cos(kQ − nt + φ m) with $$|n/k - P^ * | 〉 \Delta \upsilon$$ , is a random variable whose r.m.s. with respect to the φ m is exponentially small in the parameter $$\varepsilon = A/\Delta \upsilon ^{3/2}$$ . Using this result, a rationale is provided showing that the statistical properties of the dynamics defined by H, and of the reduced dynamics including at each time t only the terms in H such that $$|m - p(t)| \leqslant \alpha A^{2/3}$$ , can be made arbitrarily close by increasing α. For practical purposes α close to 5 is enough, as confirmed numerically. The reduced dynamics being nondeterministic, it is thus analytically shown, without using the random-phase approximation, that the statistical properties of a chaotic Hamiltonian dynamics can be made arbitrarily close to that of a stochastic dynamics. An appropriate rescaling of momentum and time shows that the statistical properties of the dynamics defined by H can be considered as independent of A, on a finite time interval, for A large. The way these results could generalize to a wider class of Hamiltonians is indicated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1023092526620
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  • 20
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    A new dromion solution of the Davey-Stewartson equation (1996)
    Springer
    Il nuovo cimento della Società Italiana di Fisica 18 (1996), S. 765-770 
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    ISSN: 0392-6737
    Keywords: Solitons ; chaos ; Solitons ; BGK modes
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Summary A new dromion solution is obtained for a (2+1)-dimensional integrable model: the Davey-Stewartson equation. Some interesting questions which emerge in the procedure of getting the solution are also discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02457363
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  • 21
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    Relaxation Oscillation Enhancement and Coherence Collapse in Semiconductor Lasers with Optical Feedback (1999)
    Springer
    Optical review 6 (1999), S. 365-371 
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    ISSN: 1349-9432
    Keywords: semiconductor laser ; optical feedback ; chaos ; relaxation oscillation ; coherence collapse
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The dynamic characteristics of a semiconductor laser with optical feedback are strongly dependent on the injection current and the reflectivity and position of the external feedback reflector. We investigated the relaxation oscillation enhancement and coherence collapse state of the laser oscillation based on the laser rate equations. It is well known that laser output power jumps with increase of the injection current due to external mode transition. But here for the first we time demonstrate the existence of a chaotic scenario within successive laser power jumps. The results calculated by numerical simulations based on the rate equations are compared with those of the experiments and good coincidence between them is found.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s10043-999-0365-0
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  • 22
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    Parametric open-plus-closed-loop control of chaos in continuous dynamical systems (1997)
    Springer
    Acta mechanica solida Sinica 10 (1997), S. 316-321 
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    ISSN: 0894-9166
    Keywords: continuous dynamical system ; chaos ; parametric open-plus-closed-loop control ; the Lorenz model
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Abstract This paper presents a parametric open-plus-closed-loop control approach to controlling chaos in continuous dynamical systems. As an example, chaos in the Lorenz model is controlled to demonstrate its application. Finally, the relations between the parametric open-plus-closed-loop control and the former control methods, such as the open-plus-closed-loop control and the parametric entrainment control, are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02207795
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  • 23
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    Global bifurcation and chaos in a van der Pol-Duffing-Mathiue system with a single-well potential oscillator (1997)
    Springer
    Acta mechanica solida Sinica 10 (1997), S. 262-275 
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    ISSN: 0894-9166
    Keywords: global bifurcation ; chaos ; nonlinear vibration ; basin ; fractal
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Abstract The global bifurcation and chaos are investigated in this paper for a van der Pol-Duffing-Mathieu system with a single-well potential oscillator by means of nonlinear dynamics. The autonomous system corresponding to the system under discussion is analytically studied to draw all global bifurcation diagrams in every parameter space. These diagrams are called basic bifurcation ones. Then fixing parameter in every space and taking the parametrically excited amplitude as a bifurcation parameter, we can observe how to evolve from a basic bifurcation diagram to a chaos pattern in terms of numerical methods. The results are sufficient to show that the system has distinct dynamic behavior. Finally, the properties of the basins of attraction are observed and the appearance of fractal basin boundaries heralding the onset of a loss of structural integrity is noted in order to consider how to control the extent and the rate of the erosion in the next paper.
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    URL: http://dx.doi.org/10.1007/BF02208076
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    On hydrodynamic instabilities, chaos and phase transition (1996)
    Springer
    Acta mechanica Sinica 12 (1996), S. 1-14 
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    ISSN: 1614-3116
    Keywords: ellipticity ; hydrodynamic instabilities ; chaos ; phase transition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Abstract Ellipticity as the underlying mechanism for instabilities of physical systems is highlighted in the study of model nonlinear evolution equations with dissipation and the study of phase transition in Van der Waals fluid. Interesting results include spiky solutions, chaotic behavior in the context of partial differential equations, as well as the nucleation process due to ellipticity in phase transition.
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    URL: http://dx.doi.org/10.1007/BF02486757
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    Global bifurcations and chaos in a Van der Pol-Duffing-Mathieu system with three-well potential oscillator (1995)
    Springer
    Acta mechanica Sinica 11 (1995), S. 357-372 
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    ISSN: 1614-3116
    Keywords: semi-analytical and semi-numerical method ; global bifurcations ; chaos ; van del Pol-Duffing-Mathieu system
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Abstract Semi-analytical and semi-numerical method is used to investigate the global bifurcations and chaos in the nonlinear system of a Van der Pol-Duffing-Mathieu oscillator. Semi-analytical and semi-numerical method means that the autonomous system, called Van der Pol-Duffing system, is analytically studied to draw all global bifurcations diagrams in parameter space. These diagrams are called basic bifurcation diagrams. Then fixing parameter in every space and taking parametrically excited amplitude as a bifurcation parameter, we can observe the evolution from a basic bifurcation diagram to chaotic pattern by numerical methods.
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    URL: http://dx.doi.org/10.1007/BF02488843
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    Chaos of liquid surface waves in a vessel under vertical excitation with slowly modulated amplitude (1997)
    Springer
    Acta mechanica Sinica 13 (1997), S. 106-112 
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    ISSN: 1614-3116
    Keywords: faraday experiment ; chaos ; resonant interaction
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Abstract Free surface waves in a cylinder of liquid under vertical excitation with slowly modulated amplitude are investigated in the current paper. It is shown by both theoretical analysis and numerical simulation that chaos may occur even for a single mode with modulation which can be used to explain Gollub and Meyer's experiment. The implied resonant mechanism accounting for this phenomenon is further elucidated.
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    URL: http://dx.doi.org/10.1007/BF02487916
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    Steady state motions of shallow arch under periodic force with 1∶2 internal resonance on the plane of physical parameters (1998)
    Springer
    Applied mathematics and mechanics 19 (1998), S. 625-635 
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    ISSN: 1573-2754
    Keywords: shallow arch ; internal resonance ; steady state motion ; bifurcation ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Mathematics , Physics
    Notes: Abstract The bifurcation dynamics of shallow arch which possesses initial deflection under periodic excitation for the case of 1∶2 internal resonance is studied in this paper. The whole parametric plane is divided into several different regions according to the types of motions; then the distribution of steady state motions of shallow arch on the plane of physical parameters is obtained. Combining with numerical method, the dynamics of the system in different regions, especially in the Hopf bifurcation region, is studied in detail. The rule of the mode interaction and the route to chaos of the system is also analysed at the end.
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    URL: http://dx.doi.org/10.1007/BF02452370
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    The study on the chaotic motion of a nonlinear dynamic system (1999)
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    Applied mathematics and mechanics 20 (1999), S. 830-836 
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    ISSN: 1573-2754
    Keywords: chaos ; Melnikov method ; Poincaré map ; phase portrait ; time-displacement diagram ; O343.5
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Mathematics , Physics
    Notes: Abstract In this paper, the system of the forced vibration $$\ddot T - \lambda _1 T + \lambda _2 T^2 + \lambda _3 T^3 = \varepsilon \left( {g\cos \omega t - \varepsilon '\dot T} \right)$$ is discussed, which contains square and cubic items. The critical condition that the system enters chaotic states is given by the Melnikov method. By Poincaré map, phase portrait and time-displacement history diagram, whether the chaos occurs is determined.
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    URL: http://dx.doi.org/10.1007/BF02452482
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    On a Physically Realistic Fast Dynamo (1998)
    Springer
    Studia geophysica et geodaetica 42 (1998), S. 335-342 
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    ISSN: 1573-1626
    Keywords: fast dynamo ; rotating convection ; chaos ; MHD
    Source: Springer Online Journal Archives 1860-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences , Physics
    Notes: Abstract As a step towards a physically realistic model of a fast dynamo, we study numerically a kinematic dynamo driven by convection in a rapidly rotating cylindrical annulus. Convection maintains the quasi-geostrophic balance whilst developing more complicated time-dependence as the Rayleigh number is increased. We incorporate the effects of Ekman suction and investigate dynamo action resulting from a chaotic flow obtained in this manner. We examine the growth rate as a function of magnetic Prandtl number Pm, which is proportional to the magnetic Reynolds number. Even for the largest value of Pm considered, a clearly identifiable asymptotic behaviour is not established. Nevertheless the available evidence strongly suggests a fast dynamo process.
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    URL: http://dx.doi.org/10.1023/A:1023360821539
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    Control of the Lorenz chaos by the exact linearization (1998)
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    Applied mathematics and mechanics 19 (1998), S. 67-73 
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    ISSN: 1573-2754
    Keywords: Reyleigh number ; Lorenz system ; chaos
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    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Mathematics , Physics
    Notes: Abstract Controlling chaos in the Lorenz system with a controllable Rayleigh number is investigated by the state space exact linearization method. Based on proving the exact linearizability, the nonlinear feedback is utilized to design the transformation changing the original chaotic system into a linear controllable one so that the control is realized. Numerical examples of control are presented.
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    The double mode model of the chaotic motion for a large deflection plate (1999)
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    Applied mathematics and mechanics 20 (1999), S. 360-364 
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    ISSN: 1573-2754
    Keywords: buckled plate ; chaos ; Poincaré section
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Mathematics , Physics
    Notes: Abstract The primary aim of this paper is to study the chaotic motion of a large deflection plate. Considered here is a buckled plate, which is simply supported and subjected to a lateral harmonic excitation. At first, the partial differential equation governing the transverse vibration of the plate is derived. Then, by means of the Galerkin approach, the partial differential equation is simplified into a set of two ordinary differential equations. It is proved that the double mode model is identical with the single mode model. The Melnikov method is used to give the approximate excitation thresholds for the occurrence of the chaotic vibration. Finally numerical computation is carried out.
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    Low-Dimensional Dynamics in Sensory Biology 1: Thermally Sensitive Electroreceptors of the Catfish (1997)
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    Journal of computational neuroscience 4 (1997), S. 335-347 
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    ISSN: 1573-6873
    Keywords: electroreceptor ; catfish ; chaos ; unstable orbit ; periodic orbit ; sensory oscillator ; noise ; random process
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Medicine , Physics
    Notes: Abstract We report the results of a search for evidence of periodic unstableorbits in the electroreceptors of the catfish. The function of thesereceptor organs is to sense weak external electric fields. Inaddition, they respond to the ambient temperature and to the ioniccomposition of the water. These quantities are encoded by receptorsthat make use of an internal oscillator operating at the level of themembrane potential. If such oscillators have three or more degreesof freedom, and at least one of which also exhibits a nonlinearity,they are potentially capable of chaotic dynamics. By detecting theexistence of stable and unstable periodic orbits, we demonstratebifurcations between noisy stable and chaotic behavior using theambient temperature as a parameter. We suggest that the techniquedeveloped herein be regarded as an additional tool for the analysisof data in sensory biology and thus can be potentially useful instudies of functional responses to external stimuli. We speculatethat the appearance of unstable orbits may be indicative of a stateof heightened sensory awareness by the animal.
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    Analysis of seismic dynamical systems (1998)
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    Journal of seismology 2 (1998), S. 159-171 
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    ISSN: 1573-157X
    Keywords: chaos ; dynamical systems ; Hurst exponent ; stochasticprocess ; time series
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Physics
    Notes: Abstract A procedure is presented for the analysis of complex stationary time series for which the Fourier power spectra reveals broadband noise or broadened pulses. We first determine the Hurst exponent from which we may know whether the time series under study is mainly random or if the data points present correlations. If the data are correlated, a chaotic analysis will reveal whether they may be interpreted as a low dimensional nonlinear system (defined by a low correlation dimension and a finite and positive Kolmogorov entropy and largest positive Lyapunov exponent) or as a stochastic process. We have studied three kind of temporal series: inter-event time series of infrasonic pulses recorded at Stromboli volcano, and, S-coda waves and microseisms, that have been recorded at the eastern Pyrenees. Results show that microseisms and Coda waves can be modeled as a low dimensional deterministic system, Correlation dimensions 2.3, 3.2, respectively. At the contrary infrasonic has resulted stochastic. This chaotic character can be attributed to the medium properties. Coda waves with scattering through a fractal distribution of scatters or to multiple reflection inside resonators (for example sedimentary basins) and microseisms as a propagation of wave guide of variable cross section which have the same temporal characteristics as a nonlinear forced oscillator.
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    Source characterisation of low frequency events at Stromboli and Vulcano Islands (Isole Eolie Italy) (1999)
    Springer
    Journal of seismology 3 (1999), S. 393-408 
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    ISSN: 1573-157X
    Keywords: cepstrum ; chaos ; correlation dimension ; nonlinear oscillators
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    Topics: Geosciences , Physics
    Notes: Abstract The discussion about the source of low frequency events generally wrecorded on active volcanoes is still open and needs deeper understanding of the phenomena involved in their generation. Most of the models view such a phenomenon as a source effect (oscillation of volcanic fluids in conduits or cracks), although a different explanation as a path or site effect exists. In the present paper we analysed 26 seismic signals recorded at Vulcano and 60 at Stromboli in order to put some constraints on the functional shape of the recorded signals. They evidence all the characteristics of the low frequency events. The spectral analysis reveals sharp peaks in the range 0.5–4 Hz, while the cepstra suggest that the signals are composed by a two-sine kind function. This suggestion is confirmed by the two-dimensional projections of the reconstructed phase space. The correlation dimension of the attractor is very close to 3 for both the volcanoes confirming the existence of a toroidal structure in the phase space. This implies that we can suggest as a model for the low frequency events source a physical mechanism very similar to a Duffing oscillator or any other quasiperiodic one. In particular in the cases of the analysed volcanoes we recognise two independent oscillations.
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    Chaotic dynamics of periodically forced spheroids in simple shear flow with potential application to particle separation (1995)
    Springer
    Rheologica acta 34 (1995), S. 504-511 
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    ISSN: 1435-1528
    Keywords: Periodically forced spheroids ; chaos ; particle separation
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    Topics: Chemistry and Pharmacology , Physics
    Notes: Abstract In this paper, we consider the technologically important problem of periodically forced spheroids in simple shear flow and demonstrate the existence of chaotic parametric regimes. The approach used by Strand (1989) (for the Strong Brownian limit) is inappropriate in the chaotic regimes corresponding to the weak Brownian limit. Our results also indicate a strong dependence of the solutions obtained on the aspect ratio of the spheroids. This strong dependence on the aspect ratio may be utilized to separate particles from a suspension of particles having different shapes but similar sizes.
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    URL: http://dx.doi.org/10.1007/BF00396563
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    A simple generalization of wisdom's perturbative method (1996)
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    Celestial mechanics and dynamical astronomy 64 (1996), S. 243-260 
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    ISSN: 1572-9478
    Keywords: Perturbative methods ; chaos ; resonance ; adiabatic invariant ; asteroids
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    Topics: Physics
    Notes: Abstract A simple generalization of Wisdom's perturbative method, as originally proposed by Wisdom (1985), is obtained. Any number of resonant cosines can be handled and the method can also accommodate more involved disturbing functions. Averaged trajectories are easily obtained by drawing level curves of the action. Here, the method is first tested for simple models of 3:1 and 2:1 resonant problems. Comparisons with numerical integration and surface-section curves show very good agreements.
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    Testing for scaling in natural forms and observables (1995)
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    Journal of statistical physics 81 (1995), S. 869-880 
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    ISSN: 1572-9613
    Keywords: Self-similarity ; self-affinity ; fractals ; scaling ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The general procedure of calculating fractal dimensions or other exponents is based on estimating some quantity as a function of scale and on assessing whether or not this function is a power law. This power law manifests itself in a log (quantity) versus log (scale) plot as a linear region (scaling). It has thus become the practice to estimate dimensions by the slope of some linear region in those log-log plots. When we are dealing with exact fractals (the Koch curve, for example) there are no problems. When, however, we are working with natural forms or observables, problems begin to emerge. In such cases the scaling region is subjectively estimated and often is only the result of the generic property of the quantity to increase monotonically or decrease monotonically as the scale goes to zero irrespective of the geometry of the object. Here we discuss these issues and suggest a procedure to deal with them.
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    Quantum transport and classical dynamics in open billiards (1996)
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    Journal of statistical physics 83 (1996), S. 203-214 
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    ISSN: 1572-9613
    Keywords: Quantum transport ; open billiards ; chaos ; Ericson fluctuation ; path-length spectrum
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    Topics: Physics
    Notes: Abstract We report numerical results of an investigation of quantum transport for a weakly opened integrable circle and chaotic stadium billiards with a pair of conducting leads. While the statistics of spacings of resonance energies commonly follow the Wigner (GOE)-like distribution, the electric conductance as a function of the Fermi wavenumber shows characteristic noisy fluctuations associated with a typical set of classical orbits unique for both billiards. The wavenumber autocorrelation for the conductance is stronger in the stadium than the circle billiard, which we show is related to the length spectrum of classical short orbits. We propose an explanation of these contrasts in terms of the effect of phase decoherence due to the underlying chaotic dynamics.
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    Chaos for the Sierpinski Carpet (1997)
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    Journal of statistical physics 88 (1997), S. 979-984 
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    ISSN: 1572-9613
    Keywords: Fractal ; chaos ; Hausdorff dimension ; thermodynamic formalism ; strong-mixing ; outer measure
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We study the chaotic behavior of the Sierpinski carpet. It is proved that this dynamical system has a chaotic set whose Hausdorff dimension equals that of the Sierpinski carpet.
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    URL: http://dx.doi.org/10.1023/B:JOSS.0000015182.90436.5b
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    Irregular Fluctuations in Uncoupled Map Lattices (1998)
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    Journal of statistical physics 90 (1998), S. 749-765 
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    ISSN: 1572-9613
    Keywords: Uncoupled logistic maps ; chaos ; fluctuations ; theoretical models ; computer simulations
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    Topics: Physics
    Notes: Abstract Analytic approximations for the spatial average and its variance are derived for a system of N uncoupled chaotic logistic maps with growth parameter r = 4. The arising nontrivial closure problem is investigated with various techniques related to the classical moment problem. A Lyapunov-like linear stability analysis is presented for the transient as well as for the fluctuation regime.
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    Note on Two Theorems in Nonequilibrium Statistical Mechanics (1999)
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    Journal of statistical physics 96 (1999), S. 1343-1349 
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    ISSN: 1572-9613
    Keywords: chaos ; fluctuation theorem ; large deviations ; chaotic hypothesis ; nonequilibrium statistical mechanics ; time reversal
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract An attempt is made to clarify the difference between a theorem derived by Evans and Searles in 1994 on the statistics of trajectories in phase space and a theorem proved by the authors in 1995 on the statistics of fluctuations on phase space trajectory segments in a nonequilibrium stationary state.
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    The correlation functions near intermittency in a one-dimensional Piecewise parabolic map (1996)
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    Journal of statistical physics 83 (1996), S. 1199-1210 
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    ISSN: 1572-9613
    Keywords: Weak intermittency ; chaos ; phase transition ; correlation function ; scaling function ; crossover behavior ; critical slowing down
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract Piecewise parabolic maps constitute a family of maps in the fully developed chaotic state and depending on a parameter that can be smoothly tuned to a weakly intermittent situation. Approximate analytic expressions are derived for the corresponding correlation functions. These expressions produce power-law decay at intermittency and a crossover from power-law decay to exponential decay below intermittency. It is shown that the scaling functions and the exponent of the power law depend on the kind of the correlations.
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    Universal homoclinic bifurcations and chaos near double resonances (1997)
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    Journal of statistical physics 86 (1997), S. 1011-1051 
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    ISSN: 1572-9613
    Keywords: Hamiltonian systems ; resonances ; homoclinic bifurcations ; chaos
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    Topics: Physics
    Notes: Abstract We study the dynamics near the intersection of a weaker and a stronger resonance inn-degree-of-freedom, nearly integrable Hamiltonian systems. For a truncated normal form we show the existence of (n−2)-dimensional hyperbolic invariant tori whose whiskers intersect inmultipulse homoclinic orbits with large splitting angles. The homoclinic obits are doubly asymptotic to solutions that “diffuse” across the weak resonance along the strong resonance. We derive a universalhomoclinic tree that describes the bifurcations of these orbits, which are shown to survive in the full normal form. We illustrate our results on a three-degree-of-freedom mechanical system.
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    Does the universe in fact contain almost no information? (1996)
    Springer
    Foundations of physics 9 (1996), S. 25-41 
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    ISSN: 1572-9524
    Keywords: complexity ; chaos ; symmetry-breaking ; decoherence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract At first sight, an accurate description of the state of the universe appears to require a mind-bogglingly large and perhaps even infinite amount of information, even if we restrict our attention to a small subsystem such as a rabbit. In this paper, it is suggested that most of this information is merely apparent, as seen from our subjective viewpoints, and that the algorithmic information content of the universe as a whole is close to zero. It is argued that if the Schrödinger equation is universally valid, then decoherence together with the standard chaotic behavior of certain non-linear systems will make the universe appear extremely complex to any self-aware subsets that happen to inhabit it now, even if it was in a quite simple state shortly after the big bang. For instance, gravitational instability would amplify the microscopic primordial density fluctuations that are required by the Heisenberg uncertainty principle into quite macroscopic inhomogeneities, forcing the current wavefunction of the universe to contain such Byzantine super-positions as our planet being in many macroscopically different places at once. Since decoherence bars us from experiencing more than one macroscopic reality, we would see seemingly complex constellations of starsetc., even if the initial wavefunction of the universe was perfectly homogeneous and isotropic.
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    Bifurcation in a nonlinear duffing system with multi-frequency external periodic forces (1998)
    Springer
    Applied mathematics and mechanics 19 (1998), S. 121-128 
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    ISSN: 1573-2754
    Keywords: nonlinear frequency ; Floquet theory ; bifurcation ; chaos ; Duffing system with multi-frequency external periodic forces
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Mathematics , Physics
    Notes: Abstract By introducing nonlinear freqyency, using Floquet theory and referring to the characteristics of the solution when it passes through the transition boundaries, all kinds of bifurcation modes and their transition boundaries of Duffing equation with two periodic excitations as well as the possible ways to chaos are studied in this paper.
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    Chaotic motion of a nonlinear thermoelastic elliptic plate (1999)
    Springer
    Applied mathematics and mechanics 20 (1999), S. 960-966 
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    ISSN: 1573-2754
    Keywords: thermoelasticity ; chaos ; Melnikov function ; Poincaré mapping ; O343.5
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Mathematics , Physics
    Notes: Abstract In the paper the nonlinear dynamic equation of a harmonically forced elliptic plate is derived, with the effects of large deflection of plate and thermoelasticity taken into account. The Melnikov function method is used to give the critical condition for chaotic motion. A demonstrative example is discussed through the Poincaré mapping, phase portrait and time history. Finally the path to chaotic motion is also discussed. Through the theoretical analysis and numerical computation some beneficial conclusions are obtained.
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