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  • Articles  (11)
  • chaos  (11)
  • Springer  (11)
  • American Geophysical Union (AGU)
  • Annual Reviews
  • 1995-1999  (11)
  • 1980-1984
  • 1935-1939
  • 1995  (11)
  • Mathematics  (11)
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  • Articles  (11)
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  • Springer  (11)
  • American Geophysical Union (AGU)
  • Annual Reviews
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  • 1995-1999  (11)
  • 1980-1984
  • 1935-1939
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  • 1
    Electronic Resource
    Electronic Resource
    Local instability and localization of attractors. From stochastic generator to Chua's systems (1995)
    Springer
    Acta applicandae mathematicae 40 (1995), S. 179-243 
    Details
    ISSN: 1572-9036
    Keywords: 58F10 ; 58F13 ; Lyapunov stability ; orbital stability ; Zhukovskij stability ; instability ; system in variations ; Lyapunov characteristic exponent ; Lyapunov function ; positive invariant set ; frequency-domain theorem ; strange attractor ; chaos ; nonlocal reduction method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper discusses the connection between various instability definitions (namely, Lyapunov instability, Poincaré or orbital instability, Zhukovskij instability) and chaotic movements. It is demonstrated that the notion of Zhukovskij instability is the most adequate for describing chaotic movements. In order to investigate this instability, a new type of linearization is offered and the connection between that and the theorems of Borg, Hartman-Olech, and Leonov is established. By means of new linearization, analytical conditions of the existence of strange attractors for impulse stochastic generators are obtained. The assumption is expressed that an analogous analytical tool may be elaborated for continuous dynamical systems describing Chua's circuits. The paper makes a first step in this direction and establishes a frequency criterion of the existence of positive invariant sets with positive Lebesgue measure for piecewise linear systems, which are unstable in every region of phase space where they are linear.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00992721
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  • 2
    Electronic Resource
    Electronic Resource
    A study of the chaotic behaviour of the elastic-plastic Euler arch (1995)
    Springer
    Nonlinear dynamics 7 (1995), S. 217-229 
    Details
    ISSN: 1573-269X
    Keywords: Nonlinear dynamics ; chaos ; elastic-plastic material ; Euler arch
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper deals with the dynamics of a truss structure, the Euler arch. The bars are made of elastic-plastic material, and the structure can exhibit large displacements. The aim of this paper is to give evidence of the possible chaotic behavior of this structure, even in the presence of a hardening plastic branch. The tools used are the diagrams of bifurcation, the measure of the dimension of the attractor, the Kolmogorov entropy, and the maximum Lyapunov exponent. This study emphasizes the sensitivity to the initial conditions by means of generalized basins of attraction.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00053709
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  • 3
    Electronic Resource
    Electronic Resource
    Parameter variation methods for cell mapping (1995)
    Springer
    Nonlinear dynamics 7 (1995), S. 273-284 
    Details
    ISSN: 1573-269X
    Keywords: Cell mapping ; continuation ; basins of attraction ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In the study of nonlinear dynamic systems, the influence of system parameters on the long term behaviour plays an important role. In this paper, parameter variation methods are presented which can be used when investigating a nonlinear dynamic system by means of simple or interpolated cell mapping. In the case of coexisting attractors, the proposed methods determine the evolution of the basin boundaries when a system parameter is varied. Application of the methods to a modified Duffing equation is performed. It is concluded that the proposed methods are very efficient and accurate.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00046303
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  • 4
    Electronic Resource
    Electronic Resource
    Lie transformation method for dynamical systems having chaotic behavior (1995)
    Springer
    Nonlinear dynamics 7 (1995), S. 403-428 
    Details
    ISSN: 1573-269X
    Keywords: Lie transformation (perturbation method) ; dynamical systems ; small divisors ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The paper persents recent developments in a singular perturbation method, known as the “Lie transformation method” for the analysis of nonlinear dynamical systems having chaotic behavior. A general approximate solution for a system of first-order differential equations having algebraic nonlinearities is introduced. Past applications to simple dynamical nonlinear models have shown that this method yields highly accurate solutions of the systems. In the present paper the capability of this method is extended to the analysis of dynamical systems having chaotic behavior: indeed, the presence of “small divisors” in the general expression of the solution suggests a modification of the method that is necessary in order to analyze nonlinear systems having chaotic behavior (indeed, even non-simple-harmonic behavior). For the case of Hamiltonian systems this is consistent with the KAM (Kolmogorov-Arnold-Moser) theory, which gives the limits of integrability for such systems; in contrast to the KAM theory, the present formulation is not limited to conservative systems. Applications to a classic aeroelastic problem (panel flutter) are also included.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00121106
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  • 5
    Electronic Resource
    Electronic Resource
    Chaos in a mapping describing elastoplastic oscillations (1995)
    Springer
    Nonlinear dynamics 8 (1995), S. 111-139 
    Details
    ISSN: 1573-269X
    Keywords: Elastoplastic oscillator ; piecewise linear map ; chaos ; Smale horseshoe ; symbolic dynamics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study the local and global dynamical behavior of a two dimensional piecewise linear map which describes the asymptotic motions of a single degree of freedom, parametrically excited, elastoplastic oscillator after it has settled down to purely elastic oscillations. We give existence and stability conditions for periodic orbits and prove that chaos, in the form of a Smale horseshoe, exists at specific, but representative, parameter values. We interpret simulations of the elastoplastic oscillator itself in the light of these results.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00045009
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  • 6
    Electronic Resource
    Electronic Resource
    Three routes to chaos for a three-degree-of-freedom articulated cylinder system subjected to annular flow and impacting on the outer pipe (1995)
    Springer
    Nonlinear dynamics 7 (1995), S. 429-450 
    Details
    ISSN: 1573-269X
    Keywords: Articulated cylinder ; chaos ; period-doubling ; intermittency ; quasi-periodicity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, the dynamics of a cantilevered articulated system of rigid cylinders interconnected by rotational springs, within a pipe containing fluid flow is studied. Although the formulation is generalized to any number of degrees-of-freedom (articulations), the present work is restricted to three-degree-of-freedom systems. The motions are considered to be planar, and the equations of motion, apart from impacting terms, are linearized. Impacting of the articulated cylinder system on the outer pipe is modelled by either a cubic spring (for analytical convenience) or, more realistically, by a trilinear spring model. The critical flow velocities, for which the system loses stability, by flutter (Hopf bifurcation) or divergence (pitchfork bifurcation) are determined by an eigenvalue analysis. Beyond these first bifurcations, it is shown that, for different values of the system parameters, chaos is obtained through three different routes as the flow is incremented: a period-doubling cascade, the quasiperiodic route, and type III intermittency. The dynamical behaviour of the system and differing routes to chaos are illustrated by phase-plane portraits, bifurcation diagrams, power spectra, Poincaré sections, and Lyapunov exponent calculations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00121107
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  • 7
    Electronic Resource
    Electronic Resource
    An experimental study of bifurcation, chaos, and dimensionality in a system forced through a bifurcation parameter (1995)
    Springer
    Nonlinear dynamics 8 (1995), S. 467-489 
    Details
    ISSN: 1573-269X
    Keywords: Experimental ; bifurcation ; chaos ; fractal dimension ; parametric excitation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An experimental study of a system that is parametrically excited through a bifurcation parameter is presented. The system consits of a lightly-damped, flexible beam which is buckled and unbuckled magnetically: it is parametrically excited by driving an electromagnet with a low-frequency sine wave. For voltage amplitudes in excess of the static bifurcation value, the beam slowly switches between the one-and two-well configurations. Experimental static and dynamic bifurcation results are presented. Static bifurcatons for the system are shown to involve a butterfly catastrophe. The dynamic bifurcation diagram, obtained with an automated data acquisition system, shows several period-doubling sequences, jump phenomena, and a chaotic region. Poincaré sections of a chaotic steady-state are obtained for various values of the driving phase, and the correlation dimension of the chaotic attractor is estimated over a large scaling region. Singular system analysis is used to demonstrate the effect of delay time on the noise level in delay-reconstructions, and to provide an independent check on the dimension estimate by directly estimating the number of independent coordinates from time series data. The correlation dimension is also estimated using the delay-reconstructed data and shown to be in good agrement with the value obtained from the Poincaré sections. The bifurcation and dimension results are used together with physical sonsiderations to derive the general form of a single-degree-of-freedom model for the experimental system.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00045709
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  • 8
    Electronic Resource
    Electronic Resource
    The analytical predictive criteria for chaos and escape in nonlinear oscillators: A survey (1995)
    Springer
    Nonlinear dynamics 7 (1995), S. 129-147 
    Details
    ISSN: 1573-269X
    Keywords: Nonlinear oscillations ; chaos ; escape ; predictive criteria
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The purpose of this paper is to provide a brief summary of the various analytical predictive criteria in order for “strange phenomena” to occur in a class of softening nonlinear oscillators, oscillators which may exhibit escape from a potential well. Implications of Melnikov's criteria are discussed first and transient chaos in the twin-well potential oscillator is illustrated. Three different heuristic criteria for steady state chaos or escape solution, proposes by F. Moon, G. Schmidt and W. Szemplińskia-Stupnicka, are then presented and compared to computer simulation results.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00053705
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  • 9
    Electronic Resource
    Electronic Resource
    Nonlinear interactions in a parametrically excited system with widely spaced frequencies (1995)
    Springer
    Nonlinear dynamics 7 (1995), S. 195-216 
    Details
    ISSN: 1573-269X
    Keywords: Energy transfer ; bifurcations ; chaos ; crises
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An investigation is presented into the transfer of energy from high- to low-frequency modes. The method of averaging is used to analyze the response of a two-degree-of-freedom system with widely spaced frequencies and cubic nonlinearities to a principal parametric resonance of the high-frequency mode. The conditions under which energy can be transferred from high- to low-frequency modes, as observed in the experiments, are determined. The interactions between the widely separated modes result in various bifurcations, the coexistence of multiple attractors, and chaotic attractors. The results show that damping may be destabilizing. The analytical results are validated by numerically solving the original system.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00053708
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  • 10
    Electronic Resource
    Electronic Resource
    Global bifurcations in externally excited two-degree-of-freedom nonlinear systems (1995)
    Springer
    Nonlinear dynamics 8 (1995), S. 85-109 
    Details
    ISSN: 1573-269X
    Keywords: Resonance ; global bifurcations ; homoclinic orbits ; Melnikov ; Silnikov ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this study we examine the global dynamics associated with a generic two-degree-of-freedom (2-DOF), coupled nonlinear system that is externally excited. The method of averaging is used to obtain the second order approximation of the response of the system in the presence of one-one internal resonance and subharmonic external resonance. This system can describe a variety of physical phenomena such as the motion of an initially deflected shallow arch, pitching vibrations in a nonlinear vibration absorber, nonlinear response of suspended cables etc. Using a perturbation method developed by Kovačič and Wiggins (1992), we show the existence of Silnikov type homoclinic orbits which may lead to chaotic behavior in this system. Here two different cases are examined and conditions are obtained for the existence of Silnikov type chaos.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00045008
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  • 11
    Electronic Resource
    Electronic Resource
    Numerical and geometrical analysis of bifurcation and chaos for an asymmetric elastic nonlinear oscillator (1995)
    Springer
    Nonlinear dynamics 7 (1995), S. 249-272 
    Details
    ISSN: 1573-269X
    Keywords: Numerical and geometrical techniques ; local and global bifurcations ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An asymmetric nonlinear oscillator representative of the finite forced dynamics of a structural system with initial curvature is used as a model system to show how the combined use of numerical and geometrical analysis allows deep insight into bifurcation phenomena and chaotic behaviour in the light of the system global dynamics. Numerical techniques are used to calculate fixed points of the response and bifurcation diagrams, to identify chaotic attractors, and to obtain basins of attraction of coexisting solutions. Geometrical analysis in control-phase portraits of the invariant manifolds of the direct and inverse saddles corresponding to unstable periodic motions is performed systematically in order to understand the global attractor structure and the attractor and basin bifurcations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00053711
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