Universality of power spectra of a dynamical system with an infinite sequence of period-doubling bifurcations

Abstract In this paper, a single-degree-of-freedom magnetic levitation dynamic system, whose spring is composed of a magnetic repulsive force, is numerically analyzed. The numerical results indicate that a body levitated by magnetic force shows many kinds of vibrations upon adjusting the system parameters (viz., damping, excitation amplitude and excitation frequency) when the system is excited by the harmonically moving base. For a suitable combination of parameters, an aperiodic vibration occurs after a sequence of period-doubling bifurcations. Typical aperiodic vibrations that occurred after period-doubling bifurcations from several initial states are identified as chaotic vibration and classified into two groups by examining their power spectra, Poincare maps, fractal dimension analyses, etc.

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FOURTH-ORDER NEARLY-SYMMETRIC CNNS EXHIBITING COMPLEX DYNAMICS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127405012867 ◽

2005 ◽

Vol 15 (05) ◽

pp. 1579-1587 ◽

Cited By ~ 10

Author(s):

M. DI MARCO ◽

M. FORTI ◽

M. GRAZZINI ◽

L. PANCIONI

Keyword(s):

Dynamical System ◽

Neural Networks ◽

Fourth Order ◽

Complex Dynamics ◽

Period Doubling ◽

Cellular Neural Networks ◽

Parameter Family ◽

Relative Errors ◽

Period Doubling Bifurcations

In this paper, the possible presence of complex dynamics in nearly-symmetric standard Cellular Neural Networks (CNNs), is investigated. A one-parameter family of fourth-order CNNs is presented, which exhibits a cascade of period-doubling bifurcations leading to the birth of a complex attractor, close to some nominal symmetric CNN. Different from previous work on this topic, the bifurcations and complex dynamics are obtained for small relative errors with respect to the nominal interconnections. The fourth-order CNNs have negative (inhibitory) interconnections between distinct neurons, and are designed by a variant of a technique proposed by Smale to embed a given dynamical system within a competitive dynamical system of larger order.

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Period Doubling and Chaos in Unsymmetric Structures Under Parametric Excitation

Journal of Applied Mechanics ◽

10.1115/1.3176195 ◽

1989 ◽

Vol 56 (4) ◽

pp. 947-952 ◽

Cited By ~ 33

Author(s):

W. Szemplin´ska-Stupnicka ◽

R. H. Plaut ◽

J.-C. Hsieh

Keyword(s):

Limit Cycles ◽

Nonlinear Oscillations ◽

Chaotic Behavior ◽

Excitation Frequency ◽

Power Spectra ◽

Period Doubling ◽

Analytical Techniques ◽

Excited System ◽

Period Doubling Bifurcations ◽

Quadratic And Cubic Nonlinearities

Nonlinear oscillations of a single-degree-of-freedom, parametrically-excited system are considered. The stiffness involves quadratic and cubic nonlinearities and models a shallow arch or buckled mechanism. The excitation frequency is assumed to be close to twice the natural frequency of the system. Numerical integration is used to obtain phase plane portraits, power spectra, and Poincare´ maps for large-time motions. Period-doubling bifurcations and several types of limit cycles and chaotic behavior are observed. Approximate analytical techniques are applied to analyze some of the limit cycles and transitions of behavior. The results are used to estimate the parameter region in which chaos may occur.

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THE RESPONSE OF A NONLINEAR SYSTEM WITH A NONSEMISIMPLE ONE-TO-ONE RESONANCE TO A COMBINATION PARAMETRIC RESONANCE

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127495000739 ◽

1995 ◽

Vol 05 (04) ◽

pp. 971-982 ◽

Cited By ~ 6

Author(s):

C. CHIN ◽

A. H. NAYFEH ◽

D. T. MOOK

Keyword(s):

Multiple Scales ◽

Nonlinear Partial Differential Equation ◽

Power Spectra ◽

Period Doubling ◽

Supersonic Stream ◽

Equilibrium Solutions ◽

Shooting Technique ◽

First Order ◽

Period Doubling Bifurcations ◽

Combination Parametric Resonance

The Galerkin procedure is used to discretize the nonlinear partial differential equation and boundary conditions governing the flutter of a simply supported panel in a supersonic stream. These equations have repeated natural frequencies at the onset of flutter. The method of multiple scales is used to derive five first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the excited modes. Then, the modulation equations are used to calculate the equilibrium solutions and their stability, and hence to identify the excitation parameters that suppress flutter and those that lead to complex motions. A combination of a shooting technique and Floquet theory is used to calculate limit cycles and their stability. The numerical results indicate the existence of a sequence of period-doubling bifurcations that culminates in chaos. The complex motions are characterized by using phase planes, power spectra, Lyapunov exponents, and dimensions.

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Can short time delays influence the variability of the solar cycle?

Proceedings of the International Astronomical Union ◽

10.1017/s1743921311017716 ◽

2010 ◽

Vol 6 (S271) ◽

pp. 288-296

Author(s):

Laurène Jouve ◽

Michael R. E. Proctor ◽

Geoffroy Lesur

Keyword(s):

Solar Cycle ◽

Convection Zone ◽

Mean Field ◽

Period Doubling ◽

Temporal Modulation ◽

Solar Convection Zone ◽

Flux Tubes ◽

Solar Convection ◽

Short Time ◽

Period Doubling Bifurcations

AbstractWe present the effects of introducing results of 3D MHD simulations of buoyant magnetic fields in the solar convection zone in 2D mean-field Babcock-Leighton models. In particular, we take into account the time delay introduced by the rise time of the toroidal structures from the base of the convection zone to the solar surface. We find that the delays produce large temporal modulation of the cycle amplitude even when strong and thus rapidly rising flux tubes are considered. The study of a reduced model reveals that aperiodic modulations of the solar cycle appear after a sequence of period doubling bifurcations typical of non-linear systems. We also discuss the memory of such systems and the conclusions which may be drawn concerning the actual solar cycle variability.

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Bifurcations and Chaos in a Minimal Neural Network: Forced Coupled Neurons

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127421501479 ◽

2021 ◽

Vol 31 (10) ◽

pp. 2150147

Author(s):

Yo Horikawa

Keyword(s):

Periodic Solution ◽

Periodic Solutions ◽

Period Doubling ◽

Output Function ◽

Piecewise Constant ◽

Border Collision Bifurcations ◽

Saddle Node ◽

Periodic Input ◽

Symmetric Periodic Solution ◽

Period Doubling Bifurcations

The bifurcations and chaos in a system of two coupled sigmoidal neurons with periodic input are revisited. The system has no self-coupling and no inherent limit cycles in contrast to the previous studies and shows simple bifurcations qualitatively different from the previous results. A symmetric periodic solution generated by the periodic input underdoes a pitchfork bifurcation so that a pair of asymmetric periodic solutions is generated. A chaotic attractor is generated through a cascade of period-doubling bifurcations of the asymmetric periodic solutions. However, a symmetric periodic solution repeats saddle-node bifurcations many times and the bifurcations of periodic solutions become complicated as the output gain of neurons is increasing. Then, the analysis of border collision bifurcations is carried out by using a piecewise constant output function of neurons and a rectangular wave as periodic input. The saddle-node, the pitchfork and the period-doubling bifurcations in the coupled sigmoidal neurons are replaced by various kinds of border collision bifurcations in the coupled piecewise constant neurons. Qualitatively the same structure of the bifurcations of periodic solutions in the coupled sigmoidal neurons is derived analytically. Further, it is shown that another period-doubling route to chaos exists when the output function of neurons is asymmetric.

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Chaotic dynamics of a pulse modulated control system for a heating unit

MATEC Web of Conferences ◽

10.1051/matecconf/201822402055 ◽

2018 ◽

Vol 224 ◽

pp. 02055

Author(s):

Yuriy A. Gol’tsov ◽

Alexander S. Kizhuk ◽

Vasiliy G. Rubanov

Keyword(s):

Control System ◽

Differential Equations ◽

Chaotic Dynamics ◽

Period Doubling ◽

Classical Period ◽

Nonlinear Phenomena ◽

Pulse Control ◽

Heating Unit ◽

Border Collision Bifurcation ◽

Period Doubling Bifurcations

The dynamic modes and bifurcations in a pulse control system of a heating unit, the condition of which is described through differential equations with discontinuous right–hand sides, have been studied. It has been shown that the system under research can demonstrate a great variety of nonlinear phenomena and bifurcation transitions, such as quasiperiodicity, multistable behaviour, chaotization of oscillations through a classical period–doubling bifurcations cascade and border–collision bifurcation.

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