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首页> 外文会议>Space Visions Congress; 20070426-28; Cocoa,FL(US) >Universal Law for the Transition from Chaos to Periodicity in Nonlinear Physical Systems
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Universal Law for the Transition from Chaos to Periodicity in Nonlinear Physical Systems

机译：非线性物理系统中从混沌到周期性过渡的通用定律

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This paper investigates the Chaos phenomena in nonlinear physical systems described by differential equations. A prototypic system is the Duffing oscillator, described by the nonlinear second order differential equation which presents a mathematical model of the motion performed by a plane pendulum under a periodic external force. By using numerical and phase space analysis, the transition from periodic to chaotic behavior (and vice versa) is analyzed. By changing the damping parameter k, the transition to chaos through the bifurcations of limit cycles is demonstrated. Numerical results show that after four successful bifurcations, the 16-cycle unexpectedly exchanges with a stable 3-cycle, which further bifurcates to a 6- and 12-cycle, until the chaotic strange at-tractor is reached. Further decrease of the damping parameter provides the transition from chaos to odd periodic limit cycles. The stable 9-, 7- and 5-periodic limit cycles successfully lead the motion, the latter bifurcates to 10-cycle, and further to 15-cycle, which again leads to a chaotic strange attractor. Finally, for small values of the damping parameter, a stable 1-limit cycle emerges from the chaos.
机译：本文研究了用微分方程描述的非线性物理系统中的混沌现象。原型系统是Duffing振荡器，由非线性二阶微分方程描述，该方程表示平面摆在周期性外力作用下进行的运动的数学模型。通过使用数值和相空间分析，可以分析从周期性到混沌行为的转变（反之亦然）。通过改变阻尼参数k，可以证明通过极限循环的分叉过渡到混沌。数值结果表明，在成功地分叉了四个周期之后，16个周期意外地与一个稳定的3个周期进行交换，该3个周期又进一步分成了6个周期和12个周期，直到达到混沌的奇异吸引子。阻尼参数的进一步减小提供了从混沌到奇数周期极限循环的过渡。稳定的9、7和5个周期极限周期成功地导致了运动，后者分叉成10个周期，进而达到了15个周期，这又导致了一个混沌的奇怪吸引子。最后，对于较小的阻尼参数值，会从混乱中出现稳定的1-极限周期。

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