Complex Route to Chaos in Velocity Driven Atoms

Abstract The Duffing driven, damped, “softening” oscillator has been analyzed for transition through period doubling route to chaos. The forcing frequency and amplitude have been varied in time (constant sweep). The stationary 2T, 4T… chaos regions have been determined and used as the starting conditions for nonstationary regimes, consisting of the transition along the Ω(t)=Ω0±α2t,f=const., Ω-line, and along the E-line: Ω(t)=Ω0±α2t;f(t)=f0∓α2t. The results are new, revealing, puzzling and complex. The nonstationary penetration phenomena (delay, memory) has been observed for a single and two-control nonstationary parameters. The rate of penetrations tends to zero with increasing sweeps, delaying thus the nonstationary chaos relative to the stationary chaos by a constant value. A bifurcation discontinuity has been uncovered at the stationary 2T bifurcation: the 2T bifurcation discontinuity drops from the upper branches of (a, Ω) or (a, f) curves to their lower branches. The bifurcation drops occur at the different control parameter values from the response x(t) discontinuities. The stationary bifurcation discontinuities are annihilated in the nonstationary bifurcation cascade to chaos — they reside entirely on the upper or lower nonstationary branches. A puzzling drop (jump) of the chaotic bifurcation bands has been observed for reversed sweeps. Extreme sensitivity of the nonstationary bifurcations to the starting conditions manifests itself in the flip-flop (mirror image) phenomena. The knowledge of the bifurcations allows for accurate reconstruction of the spatial system itself. The results obtained may model mathematically a number of engineering and physical systems.

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Route to Chaos in a CO2 Laser with Injected Signal

Coherence and Quantum Optics V ◽

10.1007/978-1-4757-0605-5_179 ◽

1984 ◽

pp. 1227-1231 ◽

Cited By ~ 4

Author(s):

F. T. Arecchi ◽

G. L. Lippi ◽

G. Puccioni ◽

J. Tredicce

Keyword(s):

Co2 Laser ◽

Route To Chaos

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Route to chaos and some properties in the boundary crisis of a generalized logistic mapping

Physica A Statistical Mechanics and its Applications ◽

10.1016/j.physa.2017.05.074 ◽

2017 ◽

Vol 486 ◽

pp. 674-680 ◽

Cited By ~ 6

Author(s):

Diogo Ricardo da Costa ◽

Rene O. Medrano-T ◽

Edson Denis Leonel

Keyword(s):

Logistic Mapping ◽

Route To Chaos ◽

Boundary Crisis

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Dynamic Behavior of Geared Rotors

Volume 4: Manufacturing Materials and Metallurgy; Ceramics; Structures and Dynamics; Controls, Diagnostics and Instrumentation; Education; IGTI Scholar Award ◽

10.1115/97-gt-187 ◽

1997 ◽

Author(s):

T. N. Shiau ◽

J. S. Rao ◽

J. R. Chang ◽

Siu-Tong Choi

Keyword(s):

Dynamic Behavior ◽

Chaotic Behavior ◽

Squeeze Film ◽

Rotor Systems ◽

Lateral Vibrations ◽

Squeeze Film Dampers ◽

Torsional Excitation ◽

Route To Chaos

This paper is concerned with the dynamic behavior of geared rotor systems supported by squeeze film dampers, wherein coupled bending torsion vibrations occur. Considering the imbalance forces and gravity, it is shown that geared rotors exhibit chaotic behavior due to non linearity of damper forces. The route to chaos in such systems is established. In geared rotor systems, it is shown that torsional excitation can induce lateral vibrations. It is shown that squeeze film dampers can suppress large amplitudes of whirl arising out of torsional excitation.

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The Chaos Dynamic of Multiproduct Cournot Duopoly Game with Managerial Delegation

Discrete Dynamics in Nature and Society ◽

10.1155/2014/206961 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10 ◽

Cited By ~ 6

Author(s):

Fang Wu ◽

Junhai Ma

Keyword(s):

Flip Bifurcation ◽

Cournot Duopoly ◽

Oligopoly Theory ◽

Output Fluctuations ◽

Duopoly Model ◽

High Volatility ◽

Route To Chaos ◽

Barrier To Entry ◽

Intermittent Chaos ◽

Consumer Tastes

Although oligopoly theory is generally concerned with the single-product firm, what is true in the real word is that most of the firms offer multiproducts rather than single products in order to obtain cost-saving advantages, cater for the diversity of consumer tastes, and provide a barrier to entry. We develop a dynamical multiproduct Cournot duopoly model in discrete time, where each firm has an owner who delegates the output decision to a manager. The principle of decision-making is bounded rational. And each firm has a nonlinear total cost function due to the multiproduct framework. The Cournot Nash equilibrium and the local stability are investigated. The tangential bifurcation and intermittent chaos are reported by numerical simulations. The results show that high output adjustment speed can lead to output fluctuations which are characterized by phases of low volatility with small output changes and phases of high volatility with large output changes. The intermittent route to chaos of Flip bifurcation and another intermittent route of Flip bifurcation which contains Hopf bifurcation can exist in the system. The study can improve our understanding of intermittent chaos frequently observed in oligopoly economy.

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