Statistical dynamics at critical bifurcationsin
Duffing-van der Pol oscillator

V CHINNATHAMBI and S RAJASEKAR^*

Department of Physics, AKGS Arts College, Srivaikundam 628 619, India
^*Department of Physics, Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli 627 012, India

Abstract:

We study the characteristic features of certain statistical quantities near critical bifurcations such as onset of chaos, sudden widening and band-merging of chaotic attractor and intermittency in a periodically driven Duffing-van der Pol oscillator. At the onset of chaos the variance of local expansion rate is found to exhibit a self-similar pattern. For all chaotic attractors the variance $\sigma_{n}(q)$ of fluctuations of coarse-grained local expansion rates of nearby orbits has a single peak. However, multiple peaks are found just before and just after the critical bifurcations. On the other hand, $\sigma_{n}(q)$associated with the coarse-grained state variable is zero far from the bifurcations. The height of the peak of $\sigma_{n}(q)$ is found to increase as the control parameter approached the bifurcation point. It is maximum at the bifurcation point. Power-law variation of maximal Lyapunov exponent and the mean value of the state variable x is observed near sudden widening and intermittency bifurcations while linear variation is seen near band-merging bifurcation. The standard deviation of local Lyapunov exponent $\Lambda(X,L)$ and the local mean value $\overline{x}(L)$ of the coordinate x calculated after every L time steps are found to approach zero in the limit $L\to \infty$ as $L^{-\beta}$. $\beta$ is sensitive to the values of control parameters. Further weak and strong chaos are characterized using the probability distribution of a k-step difference quantity $\Delta x_{k} = x_{i+k} - x_{i}$.

keywords : Duffing-van der Pol oscillator; coarse-grained local expansion rates; coarse-grained state variable; weak and strong chaos.

pacs : 05.45

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