Coexisting attractors in periodically modulated logistic maps.

We consider the logistic map wherein the nonlinearity parameter is periodically modulated. For low periods, there is multistability, namely two or more distinct dynamical attractors coexist. The case of period 2 is treated in detail, and it is shown how an extension of the kneading theory for one-dimensional maps can be applied in order to analyze the origin of bistability, and to demarcate the principal regions of bistability in the phase space. When the period of the modulation is increased-and here we choose periods which are the Fibonacci numbers-the measure of multistable regions decreases. The limit of quasiperiodic driving is approached in two different ways, by increasing the period and keeping the drive dichotomous, or by increasing the period and varying the modulation sinusoidally. In the former case, we find that multistability persists in small regions of the phase space, while in the latter, there is no evidence of multistability but strange nonchaotic attractors are created.

[1]  Awadhesh Prasad,et al.  Basin bifurcations in quasiperiodically forced coupled systems. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Celso Grebogi,et al.  Bifurcation rigidity , 1999 .

[3]  V N Chizhevsky.  Multistability in dynamical systems induced by weak periodic perturbations. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  J. Guckenheimer,et al.  The dynamics of density dependent population models , 1977, Journal of mathematical biology.

[5]  James E. Ferrell,et al.  Bistability in cell signaling: How to make continuous processes discontinuous, and reversible processes irreversible. , 2001, Chaos.

[6]  Ramakrishna Ramaswamy,et al.  Strange Nonchaotic attractors , 2001, Int. J. Bifurc. Chaos.

[7]  Ramakrishna Ramaswamy,et al.  Aperiodic Nonchaotic attractors, Strange and Otherwise , 2007, Int. J. Bifurc. Chaos.

[8]  Celso Grebogi,et al.  From High Dimensional Chaos to Stable Periodic Orbits: The Structure of Parameter Space , 1997 .

[9]  J. Yorke,et al.  Strange attractors that are not chaotic , 1984 .

[10]  Sanju,et al.  Quadratic map modulated by additive periodic forcing. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  F. Arecchi,et al.  Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a Q-switched gas laser , 1982 .

[12]  Nicholas C. Metropolis,et al.  On Finite Limit Sets for Transformations on the Unit Interval , 1973, J. Comb. Theory A.

[13]  Alexander N. Pisarchik,et al.  Control of multistability in a directly modulated diode laser , 2002 .

[14]  Kaneko,et al.  Fractalization of a torus as a strange nonchaotic attractor. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[15]  King,et al.  Bistable chaos. I. Unfolding the cusp. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[16]  Hinke M. Osinga,et al.  Multistability and nonsmooth bifurcations in the Quasiperiodically forced Circle Map , 2001, Int. J. Bifurc. Chaos.

[17]  Linz,et al.  Effect of additive and multiplicative noise on the first bifurcations of the logistic model. , 1986, Physical review. A, General physics.

[18]  J. Heagy,et al.  The birth of strange nonchaotic attractors , 1994 .

[19]  Ramakrishna Ramaswamy,et al.  Intermittency Route to Strange Nonchaotic Attractors , 1997 .

[20]  Jason A. C. Gallas,et al.  Dissecting shrimps: results for some one-dimensional physical models , 1994 .

[21]  Troy Shinbrot,et al.  Progress in the control of chaos , 1995 .

[22]  Yoshihiro Yamaguchi,et al.  New type of "crisis" showing hysteresis , 1983 .

[23]  James Testa,et al.  Study of a one-dimensional map with multiple basins , 1983 .

[24]  J. Hasty,et al.  Synthetic gene network for entraining and amplifying cellular oscillations. , 2002, Physical review letters.

[25]  M Laurent,et al.  Multistability: a major means of differentiation and evolution in biological systems. , 1999, Trends in biochemical sciences.

[26]  Zhang Yi,et al.  Multistability Analysis for Recurrent Neural Networks with Unsaturating Piecewise Linear Transfer Functions , 2003, Neural Computation.

[27]  Yihong Du,et al.  Allee effect and bistability in a spatially heterogeneous predator-prey model , 2007 .

[28]  J. Gallas,et al.  Structure of the parameter space of the Hénon map. , 1993, Physical review letters.

[29]  Foss,et al.  Multistability and delayed recurrent loops. , 1996, Physical review letters.

[30]  M. Wortis,et al.  Iterative properties of a one-dimensional quartic map: Critical lines and tricritical behavior , 1981 .

[31]  Goswami,et al.  Annihilation of one of the coexisting attractors in a bistable system , 2000, Physical review letters.

[32]  D. A. Singer,et al.  Stable Orbits and Bifurcation of Maps of the Interval , 1978 .

[33]  Farren J. Isaacs,et al.  Computational studies of gene regulatory networks: in numero molecular biology , 2001, Nature Reviews Genetics.

[34]  Ramakrishna Ramaswamy,et al.  The phase-modulated logistic map. , 2005, Chaos.

[35]  Lai,et al.  Blowout Bifurcation Route to Strange Nonchaotic Attractors. , 1996, Physical review letters.