Mandelbrot-like sets in dynamical systems with no critical points

Endler, Antônio; Gallas, Jason A. C.

doi:10.1016/j.crma.2006.02.027

Cited by 25 publications

(18 citation statements)

References 17 publications

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“…4 illustrates islands of regular signals having the same exquisite shapes found very recently in a rather different scenario: in a discrete-time dynamical system with no critical points, i.e., in a system not obeying the Cauchy-Riemann conditions [31,32]. Such striking shapes exist abundantly in the lower portion of Fig.…”

mentioning

confidence: 91%

“…1b. Thus, semiconductor lasers open the way to investigate experimentally novel and sophisticated mathematical behaviors arising from holomorphic dynamics not ruled by critical points, so far believed to be the key players in the dynamics of complex functions [31]. In summary, chaotic phases of optically injected semiconductor lasers contain peculiar accumulation boundaries and networks formed by stable periodic solutions.…”

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confidence: 99%

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Accumulation horizons and period adding in optically injected semiconductor lasers

Bonatto

Gallas

2007

Phys. Rev. E

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We study the hierarchical structuring of islands of stable periodic oscillations inside chaotic regions in phase diagrams of single-mode semiconductor lasers with optical injection. Phase diagrams display remarkable accumulation horizons: boundaries formed by the accumulation of infinite cascades of self-similar islands of periodic solutions of ever-increasing period. Each cascade follows a specific period-adding route. The riddling of chaotic laser phases by such networks of periodic solutions may compromise applications operating with chaotic signals such as e.g. secure communications. [12] reported diagrams obtained by numerical integration of the rate equations for an optically injected semiconductor laser showing some islands of periodic laser signals embedded in a sea of chaos. These important findings raise an interesting question concerning the precise structuring of laser chaotic phases. In fact, this question is the tip of a much wider problem that we consider here.Phase diagrams for discrete-time models described by mappings are common nowadays [13,14]. But the much more difficult problem of building detailed phase diagrams for models ruled by sets of nonlinear differential equations has been much less investigated. Of course, diagrams recording complex bifurcations and providing valuable insight for a few of the lowest periods have been obtained in a number of in-depth bifurcation studies using powerful continuation methods [7,15,16,17]. However, complete diagrams, discriminating simultaneously regions of arbitrarily high periods and regions with chaotic phases, remain essentially unexplored for continuous-time autonomous models. This is the problem we attack here.Our numerical simulations revealed surprising regularities existing inside the chaotic phases of the laser. As illustrated in Figs. 1 and 2, the parameter space has wide regions characterized by chaotic solutions. These chaotic phases contain both single accumulations as well as accumulations of accumulations. More specifically, chaotic laser phases are riddled with infinite sequences of periodadding cascades, each one converging toward curves that look simple (structureless), denoted "accumulation horizons", for simplicity. One example is indicated by the arrow marked A in Fig. 2a. From a theoretical point of view, a key novelty here is that the differential equations ruling the laser are autonomous equations, i.e., they do not involve time explicitly. Thus, the remarkable organization of the parameter space reported here must originate from an intrinsic interplay between variables and parameters of the laser. We found accumulation horizons to exist abundantly also in electronic circuits, atmospheric and chemical oscillators, and several other systems [18]. To fix ideas, here we focus just on the laser case. Incidentally, we mention that accumulation cascades in semiconductor lasers have been investigated by Krauskopf and Wieczorek [17] quite recently. However, their accumulations are of a very different nature than ours [19].The laser w...

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confidence: 91%

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confidence: 99%

Accumulation horizons and period adding in optically injected semiconductor lasers

Bonatto

Gallas

2007

Phys. Rev. E

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“…in a dynamical system not obeying the Cauchy-Riemann conditions (Endler & Gallas 2006). Such striking shapes and structures exist abundantly in a continuous-time system, in the lower portion of figure 1b.…”

Section: K1þ: ð2:1bþmentioning

confidence: 99%

“…Such striking shapes and structures exist abundantly in a continuous-time system, in the lower portion of figure 1b. Thus, semiconductor lasers open the way to investigate experimentally novel and sophisticated mathematical behaviours resulting from dynamics not ruled by critical points, so far believed to be the major players in the dynamics of complex functions (Endler & Gallas 2006).…”

Section: K1þ: ð2:1bþmentioning

confidence: 99%

Accumulation boundaries: codimension-two accumulation of accumulations in phase diagrams of semiconductor lasers, electric circuits, atmospheric and chemical oscillators

Bonatto

Gallas

2007

Phil. Trans. R. Soc. A.

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We report high-resolution phase diagrams for several familiar dynamical systems described by sets of ordinary differential equations: semiconductor lasers; electric circuits; Lorenz-84 low-order atmospheric circulation model; and Rössler and chemical oscillators. All these systems contain chaotic phases with highly complicated and interesting accumulation boundaries, curves where networks of stable islands of regular oscillations with everincreasing periodicities accumulate systematically. The experimental exploration of such codimension-two boundaries characterized by the presence of infinite accumulation of accumulations is feasible with existing technology for some of these systems.

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“…In fact, after the ubiquitous shrimps, these rounded and cuspidal phases are the structures most frequently observed in flows and maps. Their detailed structure has not been studied completely so far, although some results are available [39]. Box C contains a huge number of interesting periodicity phases with rather complex structures.…”

Section: Bridges In a Laser And In The Hénon Mapmentioning

confidence: 99%

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

2013

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We report numerical evidence showing that periodic oscillations can produce unexpected and wide-ranging zigzag 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.

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Mandelbrot-like sets in dynamical systems with no critical points

Cited by 25 publications

References 17 publications

Accumulation horizons and period adding in optically injected semiconductor lasers

Accumulation horizons and period adding in optically injected semiconductor lasers

Accumulation boundaries: codimension-two accumulation of accumulations in phase diagrams of semiconductor lasers, electric circuits, atmospheric and chemical oscillators

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

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