An Average Case NP-Complete Graph Coloring Problem

NP-complete problems should be hard on some instances but those may be extremely rare. On generic instances many such problems, especially related to random graphs, have been proven easy. We show the intractability of random instances of a graph coloring problem: this graph problem is hard on average unless all NP problem under all samplable (i.e., generatable in polynomial time) distributions are easy. Worst case reductions use special gadgets and typically map instances into a negligible fraction of possible outputs. Ours must output nearly random graphs and avoid any super-polynomial distortion of probabilities.Comment: 15 page

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]

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

Deformed Statistics Kullback-Leibler Divergence Minimization within a Scaled Bregman Framework

The generalized Kullback-Leibler divergence (K-Ld) in Tsallis statistics [constrained by the additive duality of generalized statistics (dual generalized K-Ld)] is here reconciled with the theory of Bregman divergences for expectations defined by normal averages, within a measure-theoretic framework. Specifically, it is demonstrated that the dual generalized K-Ld is a scaled Bregman divergence. The Pythagorean theorem is derived from the minimum discrimination information-principle using the dual generalized K-Ld as the measure of uncertainty, with constraints defined by normal averages. The minimization of the dual generalized K-Ld, with normal averages constraints, is shown to exhibit distinctly unique features.Comment: 16 pages. Iterative corrections and expansion