On Construction of an Attainability Set in the Neighborhoodof a Periodic Attractor

This paper investigates a finance system with nonconstant elasticity of demand. First, under some conditions, the system has invariant algebraic surfaces and the analytic expressions of the surfaces are given. Furthermore, when the two surfaces coincide and become one surface, the dynamics on the surface are analyzed and a globally stable equilibrium is found. Second, by using the normal form theory, the Hopf bifurcation is studied and the approximate expression and stability of the bifurcating periodic orbit are obtained. Third, the chaotic behaviors are investigated and the route to chaos is period-doubling bifurcations. Moreover, it is found that the system has coexisting attractors, including periodic attractor and periodic attractor, chaotic attractor and chaotic attractor. With the change of parameter, the two chaotic attractors coincide and then a symmetrical chaotic attractor arises.

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Exponential periodic attractor of impulsive BAM networks with finite distributed delays

Chaos Solitons & Fractals ◽

10.1016/j.chaos.2007.04.014 ◽

2009 ◽

Vol 39 (1) ◽

pp. 373-384 ◽

Cited By ~ 18

Author(s):

Zhenkun Huang ◽

Yonghui Xia

Keyword(s):

Distributed Delays ◽

Periodic Attractor

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Phase Shift Near Natural Frequencies of Non Linear Rods

Volume 6C: 18th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc2001/vib-21588 ◽

2001 ◽

Author(s):

Ioannis T. Georgiou

Keyword(s):

Phase Shift ◽

Natural Frequencies ◽

Basin Of Attraction ◽

Continuous Phase ◽

Phase Shifting ◽

Periodic Attractor ◽

Jump Frequency ◽

Stable Attractor ◽

Configuration Changes ◽

Transient And Steady State

Abstract We study the transient and steady state dynamics of a special class of motions of forced planar rods with exact geometric nonlinearity. The attractors of these motions are separated by a finite jump at a critical forcing frequency in an attractor diagram of the undistorted configuration generated by a quasi-static frequency sweep at fixed forcing amplitude. As the frequency of the forcing passes through this critical or jump frequency, the motion (trajectory) of the undistorted configuration changes basin of attraction. For forcing frequency slightly greater than the jump frequency, the response trajectories of the undistorted configuration pass near an unstable periodic attractor and undergo continuous phase shift while approaching a stable attractor. For forcing frequency slightly smaller than the jump frequency, the response trajectories of the undistorted configuration pass near the same unstable attractor and undergo no net phase angle when landing on the stable attractor that attracts them. The phase-shifting property reveals that the frequncy at which the jump occurs is indeed a natural frequncy of the nonlinear rod.

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A Mathematical Model of Demand-Supply Dynamics with Collectability and Saturation Factors

International Journal of Bifurcation and Chaos ◽

10.1142/s021812741750016x ◽

2017 ◽

Vol 27 (01) ◽

pp. 1750016 ◽

Cited By ~ 3

Author(s):

Y. Charles Li ◽

Hong Yang

Keyword(s):

Mathematical Model ◽

Market Equilibrium ◽

Basins Of Attraction ◽

Periodic Attractor ◽

Danger Zone ◽

Small Perturbations ◽

Demand And Supply ◽

Flash Crash ◽

Period Map ◽

Irrational Exuberance

We introduce a mathematical model on the dynamics of demand and supply incorporating collectability and saturation factors. Our analysis shows that when the fluctuation of the determinants of demand and supply is strong enough, there is chaos in the demand-supply dynamics. Our numerical simulation shows that such a chaos is not an attractor (i.e. dynamics is not approaching the chaos), instead a periodic attractor (of period-3 under the Poincaré period map) exists near the chaos, and coexists with another periodic attractor (of period-1 under the Poincaré period map) near the market equilibrium. Outside the basins of attraction of the two periodic attractors, the dynamics approaches infinity indicating market irrational exuberance or flash crash. The period-3 attractor represents the product’s market cycle of growth and recession, while period-1 attractor near the market equilibrium represents the regular fluctuation of the product’s market. Thus our model captures more market phenomena besides Marshall’s market equilibrium. When the fluctuation of the determinants of demand and supply is strong enough, a three leaf danger zone exists where the basins of attraction of all attractors intertwine and fractal basin boundaries are formed. Small perturbations in the danger zone can lead to very different attractors. That is, small perturbations in the danger zone can cause the market to experience oscillation near market equilibrium, large growth and recession cycle, and irrational exuberance or flash crash.

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TRANSITION TO CHAOS IN A CURVED CARBON NANOTUBE UNDER HARMONIC EXCITATION

International Journal of Modern Physics B ◽

10.1142/s0217979212502104 ◽

2012 ◽

Vol 26 (32) ◽

pp. 1250210 ◽

Cited By ~ 2

Author(s):

YIXIANG GENG ◽

LIXIANG ZHANG

Keyword(s):

Carbon Nanotube ◽

Chaotic Behavior ◽

Numerical Study ◽

Melnikov Method ◽

Equation Of Motion ◽

Harmonic Excitation ◽

Periodic Attractor ◽

Runge Kutta Method ◽

Transition To Chaos ◽

Melnikov Criterion

The chaotic behavior of a carbon nanotube with waviness along its axis is investigated. The equation of motion involves a quadratic and cubic terms due to the curved geometry and the mid-plane stretching. Melnikov method is applied for the system, and Melnikov criterion for global homoclinic bifurcations is obtained analytically. The numerical solution of the system using a fourth-order-Runge–Kutta method confirms the analytical predictions and shows that the transition from regular to chaotic motion is often associated with increasing the energy of an oscillator. Moreover, a detailed numerical study of the periodic attractor in the period window is also carried out.

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