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  • Articles  (46)
  • chaos  (46)
  • Springer  (46)
  • American Chemical Society (ACS)
  • Cell Press
  • 1995-1999  (44)
  • 1980-1984  (2)
  • Mathematics  (45)
  • Economics  (1)
  • 1
    Electronic Resource
    Electronic Resource
    Potentialised Partial Differential Equations in Economic Geography and Spatial Economics: Multiple Dimensions and Control (1998)
    Springer
    Acta applicandae mathematicae 51 (1998), S. 1-23 
    Details
    ISSN: 1572-9036
    Keywords: partial differential equations ; potentials ; chaos ; control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The present study is a follow-up to previous recent publications in the field of theoretical economic geography and spatial economics. Earlier results are generalised and simulated in higher dimensions (in terms of variables and topological dimensions), and given possible undesirable outcomes of the process (which can behave chaotically), application of control methods to it is being studied.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1005848024409
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  • 2
    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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  • 3
    Electronic Resource
    Electronic Resource
    Global bifurcations of a periodically forced nonlinear oscillator (1984)
    Springer
    Journal of mathematical biology 21 (1984), S. 175-190 
    Details
    ISSN: 1432-1416
    Keywords: limit cycle ; phase-locking ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The effects of periodic pulsatile stimulation on a simple mathematical model of biological oscillations, called the radial isochron clock (RIC), are investigated as a function of stimulus frequency and amplitude. This system can be reduced to a two parameter, one-dimensional circle map. Numerical and topological methods are used to give a very detailed picture of the observed bifurcations over the complete range of parameters. The bifurcations are generic for a class of models which generalize the RIC.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00277669
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  • 4
    Electronic Resource
    Electronic Resource
    The dynamics of recurrent inhibition (1984)
    Springer
    Journal of mathematical biology 19 (1984), S. 211-225 
    Details
    ISSN: 1432-1416
    Keywords: lateral inhibition ; recurrent inhibition ; delayed feedback ; multiple steady states ; chaos ; epilepsy ; hippocampus ; receptor ; physiology ; difference-differential equations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A heuristic model for the dynamics of recurrent inhibition, emphasizing non-linearities arising from the stoichiometry of transmitter-receptor interactions and time delays due to finite feedback pathway transmission times, is developed and analyzed. It is demonstrated that variation in model parameters may lead to the existence of multiple steady states, and the local stability of these are analyzed as well as the occurrence of switching behaviour between them. As an example of the applicability of this model, parameters are estimated for the hippocampal mossy fibre-CA3 pyramidal cell-basket cell complex. Numerically simulated responses of this system to alterations in presynaptic drive and titration of inhibitory transmitter receptors by penicillin are presented. Numerical simulations indicate the existence of multiple bifurcations between periodic solutions, as well as the existence of bifurcations to chaotic solutions, as presynaptic drive and receptor density are varied. It is hypothesized that the model offers insight into the sequences of events recorded in single CA3 pyramidal cells following the application of penicillin, a specific inhibitory receptor blocking agent.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00277747
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  • 5
    Electronic Resource
    Electronic Resource
    The analysis of correlation integrals in terms of extremal value theory (1998)
    Springer
    Bulletin of the Brazilian Mathematical Society 29 (1998), S. 197-228 
    Details
    ISSN: 1678-7714
    Keywords: time series ; (deterministic) ; chaos ; correlation integral ; predictability ; extremal index
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we first give an overview of the methods of analysis of time series in terms of correlation integrals, which were developed for time series generated by deterministic systems. From the extremal value theory one obtains asymptotic information on the behaviour of the correlation integrals of time series generated by non-deterministic (mixing) systems. This leads to an analysis in terms of correlation integrals which is complementary to the estimation of dimension and entropy.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01237649
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  • 6
    Electronic Resource
    Electronic Resource
    Numerical approximation of homoclinic chaos (1997)
    Springer
    Numerical algorithms 14 (1997), S. 25-53 
    Details
    ISSN: 1572-9265
    Keywords: dynamical systems ; numerical methods ; homoclinic points for maps ; multihumped homoclinic orbits ; chaos ; 58F13 ; 58F15 ; 58F08
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Transversal homoclinic orbits of maps are known to generate a Cantor set on which a power of the map conjugates to the Bernoulli shift on two symbols. This conjugacy may be regarded as a coding map, which for example assigns to a homoclinic symbol sequence a point in the Cantor set that lies on a homoclinic orbit of the map with a prescribed number of humps. In this paper we develop a numerical method for evaluating the conjugacy at periodic and homoclinic symbol sequences in a systematic way. The approach combines our previous method for computing the primary homoclinic orbit with the constructive proof of Smale's theorem given by Palmer. It is shown that the resulting nonlinear systems are well conditioned uniformly with respect to the characteristic length of the symbol sequence and that Newton's method converges uniformly too when started at a proper pseudo orbit. For the homoclinic symbol sequences an error analysis is given. The method works in arbitrary dimensions and it is illustrated by examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019196426363
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  • 7
    Electronic Resource
    Electronic Resource
    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 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02452370
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  • 8
    Electronic Resource
    Electronic Resource
    The study on the chaotic motion of a nonlinear dynamic system (1999)
    Springer
    Applied mathematics and mechanics 20 (1999), S. 830-836 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02452482
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  • 9
    Electronic Resource
    Electronic Resource
    Control of the Lorenz chaos by the exact linearization (1998)
    Springer
    Applied mathematics and mechanics 19 (1998), S. 67-73 
    Details
    ISSN: 1573-2754
    Keywords: Reyleigh number ; Lorenz system ; 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 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02458982
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  • 10
    Electronic Resource
    Electronic Resource
    The double mode model of the chaotic motion for a large deflection plate (1999)
    Springer
    Applied mathematics and mechanics 20 (1999), S. 360-364 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02458561
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