The renormalization group method of Goldenfeld, Oono and their collaborators is applied to asymptotic analysis of vector fields. The method is formulated on the basis of the theory of envelopes, as was done for scalar fields. This formulation actually completes the discussion of the previous work for scalar equations. It is shown in a generic way that the method applied to equations with a bifu...

A period-doubling cascade is often seen in numerical studies of those smooth (one-parameter families of) maps for which as the parameter is varied, the map transitions from one without chaos to one with chaos. Our emphasis in this paper is on establishing the existence of such a cascade for many maps with phase space dimension 2. We use continuation methods to show the following: under certain ...

A well-known method to suppress chaos in a periodically forced chaotic system is to add a harmonic perturbation. The phase control of chaos scheme uses the phase difference between a small added harmonic perturbation and the main driving to suppress chaos, leading the system to different periodic orbits. Using the Duffing oscillator as a paradigm, we present here an in-depth study of this techn...

The asymmetrically forced, damped Duffing oscillator is introduced as a prototype model for analyzing the homoclinic tangle of symmetric dissipative systems with symmetry-breaking disturbances. Even a slight fixed asymmetry in the perturbation may cause a substantial change in the asymptotic behavior of the system, e.g., transitions from two-sided to one-sided strange attractors as the other pa...

In this paper, we study the Duffing equation with one degenerate saddle point and one external forcing and obtain the criteria of chaos of Duffing equation under periodic perturbation through Melnikov method. Numerical simulations not only show the correctness of the theoretical analysis but also exhibit the more new complex dynamical behaviors, including homoclinic bifurcation, bifurcation dia...

The Lyapunov based approach for synchronizing and identifying the unknown parameters in the Duffing system is presented in this paper. The strategy consists of proposing a slave system which has to follow asymptotically the desired Duffing system. The gains of the slave system are adjusted continually according to a convenient adaptation control law, until the measurable output error converges ...

The paper proposed a new method for the detection of QPSK carrier signal that based on double duffing oscillator to solve the problems as high attenuation and strong noise interference in the QPSK carrier communication of low voltage power line. This method has making use of the different phase of QPSK carrier signal to stimulate the double duffing oscillators to enter different states and thro...

Sustained resonance in a linear oscillator is achievable with a drive whose constant frequency matches the resonant frequency of the oscillator. But in oscillators with nonlinear restoring forces such as the pendulum, Duffing and Duffing-Van der Pol oscillator, the resonant frequency changes as the amplitude changes, so a constant frequency drive results in a beat oscillation instead of sustain...

Aiming at the feature extraction of weak periodic signal in early fault of machinery, a novel weak periodic signal detection with Duffing oscillator based on empirical mode decomposition (EMD) was presented. The chaotic character of Duffing oscillator was analyzed, and the Melnikov method of determining chaotic threshold of Duffing oscillators was discussed. The principle of weak signal detecti...