Deterministic SR in a Piecewise Linear Chaotic Map

The phenomenon of Stochastic Resonance (SR) is observed in a completely deterministic setting - with thermal noise being replaced by one-dimensional chaos. The piecewise linear map investigated in the paper shows a transition from symmetry-broken to symmetric chaos on increasing a system parameter. In the latter state, the chaotic trajectory switches between the two formerly disjoint attractors, driven by the map's inherent dynamics. This chaotic switching rate is found to `resonate' with the frequency of an externally applied periodic perturbation (multiplicative or additive). By periodically modulating the parameter at a specific frequency $\omega$, we observe the existence of resonance where the response of the system (in terms of the residence-time distribution) is maximum. This is a clear indication of SR-like behavior in a chaotic system.Comment: 6 pages LaTex, 4 figure

The travelling salesman problem on randomly diluted lattices: results for small-size systems

If one places N cities randomly on a lattice of size L, we find that the normalized optimal travel distances per city in the Euclidean and Manhattan metrics vary monotonically with the city concentration p. We have studied such optimal tours for visiting all the cities using a branch and bound algorithm, giving exact optimized tours for small system sizes (N<100). Extrapolating the results for N tending to infinity, we find that the normalized optimal travel distances per city in the Euclidean and Manhattan metrics both equal unity for p=1, and they reduce to about 0.74 and 0.94, respectively, as p tends to zero. Although the problem is trivial for p=1, it certainly reduces to the standard TSP on continuum (NP-hard problem) for p tending to zero. We did not observe any irregular behaviour at any intermediate point. The crossover from the triviality to the NP-hard problem seems to occur at p=1.Comment: 7 pages, 4 figures. Revised version with changes in text and figures (to be published in Euro. Phys. Jour. B