Abstract
The Josephson equation is investigated in detail: the existence and bifurcationsfor harmonic and subharmonic solutions under small perturbations areobtained by using second-order averaging method and subharmonic Melnikov function,and the criterion of existence for chaos is proved by Melnikov analysis; thebifurcation curves about n-subharmonic and heteroclinic orbits and the driving frequency$\omega$ effects to the forms of chaotic behaviors are given by numerical simulations.
Highlights
In this paper we consider the Josephson system x = y y = −sinx − ksin2x + β − α(cosx + 2kcos2x)y + f sinωt. (A)Where x(t) is the phase-error process; sinx+ksin2x is the Hybrid loop which represents the phase-detector characteristics; ω and f are angular frequency and amplitude of the driving current respectively; sinωt represents a sinus plus noise; f sinωt is a small sinusoidal force; −α(cosx + 2kcos2x)y + β is a characteristic of transfer functions of the ideal filter.The domain of definition of the Josephson system is the tangent bundle of the circle TS1 = R2×S1, i.e., the cylinder
The Josephson equation is investigated in detail: the existence and bifurcations for harmonic and subharmonic solutions under small perturbations are obtained by using second-order averaging method and subharmonic Melnikov function, and the criterion of existence for chaos is proved by Melnikov analysis; the bifurcation curves about n-subharmonic and heteroclinic orbits and the driving frequency ω effects to the forms of chaotic behaviors are given by numerical simulations
The Eq(A) is investigated in detail: the existence and bifurcations for harmonic and subharmonic solutions under small perturbations are obtained by using second-order averaging method and subharmonic Melnikov function, and the criterion of existence for chaos is proved by Melnikov analysis; the bifurcation curves about n-subharmonic and heteroclinic orbits are given by numerical simulations
Summary
The Josephson equation is investigated in detail: the existence and bifurcations for harmonic and subharmonic solutions under small perturbations are obtained by using second-order averaging method and subharmonic Melnikov function, and the criterion of existence for chaos is proved by Melnikov analysis; the bifurcation curves about n-subharmonic and heteroclinic orbits and the driving frequency ω effects to the forms of chaotic behaviors are given by numerical simulations. The Eq(A) is investigated in detail: the existence and bifurcations for harmonic and subharmonic solutions under small perturbations are obtained by using second-order averaging method and subharmonic Melnikov function, and the criterion of existence for chaos is proved by Melnikov analysis; the bifurcation curves about n-subharmonic and heteroclinic orbits are given by numerical simulations.
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