Different routes to chaos via strange nonchaotic attractor in a quasiperiodically forced system

This paper focusses attention on the strange nonchaotic attractors (SNA) of a quasiperiodically forced dynamical system. Several routes, including the standard ones by which the appearance of strange nonchaotic attractors takes place, are shown to be realizable in the same model over a two parameters ($f-\epsilon$) domain of the system. In particular, the transition through torus doubling to chaos via SNA, torus breaking to chaos via SNA and period doubling bifurcations of fractal torus are demonstrated with the aid of the two parameter ($f-\epsilon$) phase diagram. More interestingly, in order to approach the strange nonchaotic attractor, the existence of several new bifurcations on the torus corresponding to the novel phenomenon of torus bubbling are described. Particularly, we point out the new routes to chaos, namely, (1) two frequency quasiperiodicity $\to$ torus doubling $\to$ torus merging followed by the gradual fractalization of torus to chaos, (2) two frequency quasiperiodicity $\to$ torus doubling $\to$ wrinkling $\to$ SNA $\to$ chaos $\to$ SNA $\to$ wrinkling $\to$ inverse torus doubling $\to$ torus $\to$ torus bubbles followed by the onset of torus breaking to chaos via SNA or followed by the onset of torus doubling route to chaos via SNA. The existence of the strange nonchaotic attractor is confirmed by calculating several characterizing quantities such as Lyapunov exponents, winding numbers, power spectral measures and dimensions. The mechanism behind the various bifurcations are also briefly discussed.Comment: 12 pages, 12 figures, ReVTeX (to appear in Phys. Rev. E

Bifurcation and chaos in the double well Duffing-van der Pol oscillator: Numerical and analytical studies

The behaviour of a driven double well Duffing-van der Pol (DVP) oscillator for a specific parametric choice ($\mid \alpha \mid =\beta$) is studied. The existence of different attractors in the system parameters ($f-\omega$) domain is examined and a detailed account of various steady states for fixed damping is presented. Transition from quasiperiodic to periodic motion through chaotic oscillations is reported. The intervening chaotic regime is further shown to possess islands of phase-locked states and periodic windows (including period doubling regions), boundary crisis, all the three classes of intermittencies, and transient chaos. We also observe the existence of local-global bifurcation of intermittent catastrophe type and global bifurcation of blue-sky catastrophe type during transition from quasiperiodic to periodic solutions. Using a perturbative periodic solution, an investigation of the various forms of instablities allows one to predict Neimark instablity in the $(f-\omega)$ plane and eventually results in the approximate predictive criteria for the chaotic region.Comment: 15 pages (13 figures), RevTeX, please e-mail Lakshmanan for figures, to appear in Phys. Rev. E. (E-mail: [email protected]

The radial gradient of interplanetary radiation measured by Mariners 4 and 5

Interplanetary radiation radial gradient measured from Mariners 4 and

Constraints on First-Light Ionizing Sources from Optical Depth of the Cosmic Microwave Background

We examine the constraints on high-redshift star formation, ultraviolet and X-ray pre-ionization, and the epoch of reionization at redshift z_r, inferred from the recent WMAP-5 measurement, tau_e = 0.084 +/- 0.016, of the electron scattering optical depth of the cosmic microwave background (CMB). Half of this scattering can be accounted for by the optical depth, tau_e = 0.04-0.05, of a fully ionized intergalactic medium (IGM) at z < z_GP = 6-7, consistent with Gunn-Peterson absorption in neutral hydrogen. The required additional optical depth, Delta-tau_e = 0.03 +/- 0.02 at z > z_GP, constrains the ionizing contributions of first light sources. WMAP-5 also measured a significant increase in small-scale power, which lowers the required efficiency of star formation and ionization from mini-halos. Early massive stars (UV radiation) and black holes (X-rays) can produce a partially ionized IGM, adding to the residual electrons left from incomplete recombination. Inaccuracies in computing the ionization history, x_e(z), and degeneracies in cosmological parameters (Omega_m, Omega_b, sigma_8, n_s) add systematic uncertainty to the measurement and modeling of $\tau_e$. From the additional optical depth from sources at z > z_GP, we limit the star-formation efficiency, the rate of ionizing photon production for Pop III and Pop II stars, and the photon escape fraction, using standard histories of baryon collapse, minihalo star formation, and black-hole X-ray preionization.Comment: Greatly revised version, based on WMAP-5 results and new models. Accepted for ApJ (2008