Zig-zag networks of self-excited periodic oscillations in a tunnel diode and a fiber-ring laser.

We report numerical evidence showing that periodic oscillations can produce unexpected and wide-ranging zig-zag parameter networks embedded in chaos in the control space of nonlinear systems. Such networks interconnect shrimplike windows of stable oscillations and are illustrated here for a tunnel diode, for an erbium-doped fiber-ring laser, and for the Hénon map, a proxy of certain CO(2) lasers. Networks in maps can be studied without the need for solving differential equations. Tuning parameters along zig-zag networks allows one to continuously modify wave patterns without changing their chaotic or periodic nature. In addition, we report convenient parameter ranges where such networks can be detected experimentally.

[1]  M. Rabinovich,et al.  Stochastic oscillations in dissipative systems , 1981 .

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

[3]  Leandro Junges,et al.  Frequency and peak discontinuities in self-pulsations of a CO2 laser with feedback , 2012 .

[4]  Shen Ke,et al.  Controlling hyperchaos in erbium-doped fibre laser , 2003 .

[5]  Vassilios Kovanis,et al.  Labyrinth bifurcations in optically injected diode lasers , 2010 .

[6]  Yoshisuke Ueda,et al.  Chaotic phase similarities and recurrences in a damped-driven Duffing oscillator. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  Torsion-adding and asymptotic winding number for periodic window sequences , 2012, 1210.5883.

[8]  Eusebius J. Doedel,et al.  Hysteresis of periodic and Chaotic Passive Q-Switching Self-pulsations in a molecular Laser Model, and the Stark Effect as a codimension-2 parameter , 2012, Int. J. Bifurc. Chaos.

[9]  Jason A. C. Gallas,et al.  The Structure of Infinite Periodic and Chaotic Hub Cascades in Phase Diagrams of Simple Autonomous Flows , 2010, Int. J. Bifurc. Chaos.

[10]  Roberto Barrio,et al.  Topological changes in periodicity hubs of dissipative systems. , 2012, Physical review letters.

[11]  Marko Robnik,et al.  Shrimp-shape domains in a dissipative kicked rotator. , 2011, Chaos.

[12]  Edson D. Leonel,et al.  Parameter space for a dissipative Fermi–Ulam model , 2011 .

[13]  J. Gallas,et al.  Arithmetical signatures of the dynamics of the Hénon map. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  J. G. Freire,et al.  Stern-Brocot trees in the periodicity of mixed-mode oscillations. , 2011, Physical chemistry chemical physics : PCCP.

[15]  J. A. Wheeler,et al.  Waves at walls, corners, heights: Looking for simplicity , 1995 .

[16]  J. G. Freire,et al.  Stern-Brocot trees in cascades of mixed-mode oscillations and canards in the extended Bonhoeffer-van der Pol and the FitzHugh-Nagumo models of excitable systems , 2011 .

[17]  Cristian Bonatto,et al.  Self-similarities in the frequency-amplitude space of a loss-modulated CO2 laser. , 2005, Physical review letters.

[18]  J. Gallas,et al.  Magnetization Dynamics Under a Quasiperiodic Magnetic Field , 2012, IEEE Transactions on Magnetics.

[19]  E. Leonel,et al.  Dynamical properties for the problem of a particle in an electric field of wave packet: Low velocity and relativistic approach , 2012 .

[20]  Thorsten Pöschel,et al.  Stern-Brocot trees in spiking and bursting of sigmoidal maps , 2012 .

[21]  J. Gallas,et al.  Periodicity hub and nested spirals in the phase diagram of a simple resistive circuit. , 2008, Physical review letters.

[22]  Quantum ratchets in dissipative chaotic systems. , 2004, Physical review letters.

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

[24]  J. G. Freire,et al.  Non-Shilnikov cascades of spikes and hubs in a semiconductor laser with optoelectronic feedback. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  K. Alan Shore,et al.  Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Lasers , 2005 .

[26]  P. Glendinning,et al.  Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  Yang Senlin,et al.  Study on the method of controlling chaos in an Er-doped fiber dual-ring laser via external optical injection and shifting optical feedback light. , 2007, Chaos.

[28]  E. Doedel,et al.  Isolas of periodic passive Q-switching self-pulsations in the three-level:two-level model for a laser with a saturable absorber. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  Mikhail I. Rabinovich,et al.  Stochastic self-oscillations and turbulence , 1978 .

[30]  J Bragard,et al.  Chaotic dynamics of a magnetic nanoparticle. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  R. Meucci,et al.  Highly dissipative Hénon map behavior in the four-level model of the CO2 laser with modulated losses , 1995 .

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

[33]  Leandro Junges,et al.  Intricate routes to chaos in the Mackey-Glass delayed feedback system , 2012 .

[34]  Edward N. Lorenz,et al.  Compound windows of the Hénon-map , 2008 .

[35]  M. Heckel,et al.  Circular ratchets as transducers of vertical vibrations into rotations. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  Jason A. C. Gallas,et al.  Lyapunov exponents for a Duffing oscillator , 1995 .

[37]  E. R. Viana,et al.  High-resolution parameter space of an experimental chaotic circuit. , 2010, Chaos.

[38]  Andrey Shilnikov,et al.  Global organization of spiral structures in biparameter space of dissipative systems with Shilnikov saddle-foci. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  Liguo Luo,et al.  Chaotic behavior in erbium-doped fiber-ring lasers , 1998 .

[40]  C. Manchein,et al.  Temperature resistant optimal ratchet transport. , 2012, Physical review letters.

[41]  Jason A. C. Gallas,et al.  Mandelbrot-like sets in dynamical systems with no critical points , 2006 .

[42]  J. Gallas,et al.  Accumulation boundaries: codimension-two accumulation of accumulations in phase diagrams of semiconductor lasers, electric circuits, atmospheric and chemical oscillators , 2008, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[43]  Cristiane Stegemann,et al.  Lyapunov exponent diagrams of a 4-dimensional Chua system. , 2011, Chaos.