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 ever-increasing 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.
The aim of this work is to report a number of very high-resolution phase diagrams for a few familiar dynamical systems with the aim of stimulating experimental work to corroborate several new features discovered in them. A key novelty reported here is that chaotic phases of dynamical systems modelled by coupled sets of nonlinear ordinary differential equations contain certain parameter networks with characteristic accumulation boundaries in phase diagrams, and many levels of self-similar behaviours, as might be recognized from figures 1 and 2. Here we illustrate for a few systems the relative abundance, shape and structuring of such networks of accumulations seen to exist abundantly in phase diagrams.
Figures 1 and 2 are high-resolution phase diagrams obtained by computing the spectra of Lyapunov exponents for the standard rate equation model of the laser, defined by equations (2.1a) and (2.1b). The boundaries of the chaotic laser phases contain certain segments (as the black–yellow transition boundary indicated by the letter A in figure 2a) along which wide networks of islands of regular oscillations with increasingly higher periodicities accumulate systematically. Such accumulation boundaries, limiting curves defined by the infinite network of islands, do not seem to have been observed before, not even for the computationally much less-demanding discrete-time dynamical systems where they must certainly exist. These are truly codimension-two phenomena.
Since clean and extended domains of chaos are important for a number of applications, e.g. for secure communications (Kane & Shore 2005; Ohtsubo 2005), it seems natural to inquire about what exactly is embedded in the seas of chaos abundant in dynamical systems. This is the leitmotiv here, for different types of dynamical systems. First, we discuss a semiconductor laser with injection.
2. Case study: semiconductor laser with injection
It is well known that semiconductor lasers present a rich nonlinear phenomenology when subjected to optical injection, optical feedback or modulations (Dorn et al. 2003; Kane & Shore 2005; Ohtsubo 2005). In particular, optically injected semiconductor lasers have attracted much attention in recent years, experimentally as well as from a theoretical and numerical point of view, as summarized in a recent survey by Wieczorek et al. (2005).
As mentioned, theoretical and numerical works based on a rate equation model have revealed a complex structure of bifurcations in parameter space as a function of the injected intensity and the frequency detuning. Simpson et al. (1997) located experimentally a number of bifurcations while Wieczorek et al. (2002) compared measurements with numerical computations, obtaining a very good overall agreement between theory and experiments.
Theoretical calculations (Gao et al. 1999) and numerical simulations (Hwang & Liu 2000) predicted intricate laser behaviours, including stable solutions inside domains characterized by chaotic laser oscillations. Recently, Chlouverakis & Adams (2003) and Fordell & Lindberg (2004) reported a series of stability diagrams obtained by direct numerical integration of the rate equations for an optically injected semiconductor laser, where it is possible to identify various stability islands embedded in a sea of chaos. However, not much is presently known about the shape, abundance and structuring of such stability islands.
The single-mode semiconductor laser subjected to monochromatic optical injection is well described by rescaled rate equations for the complex laser field E=Ex+iEy and population inversion n(2.1a)(2.1b)Here, the control parameters are the intensity K of the injected field and ω, the detuning frequency. The remaining parameters are defined as usual: B=0.0295, Γ=0.0973 and α=2.6 (Wieczorek et al. 2002).
Figure 1 shows phase diagrams obtained by integrating the above equations with a standard four-order Runge–Kutta scheme with a fixed step h=0.01. The Lyapunov spectra are computed using farming (Zeni & Gallas 1995). Parameter grid points were colour codified according to the magnitude of the largest non-zero exponent: regions of negative exponents (periodic solutions) are coloured grey (black indicates zero, white the most negative values), while positive exponents (chaotic laser oscillations) are indicated in a yellow–red scale. Red indicates regions of stronger chaos, characterized by more positive exponents. The colour scale of individual phase diagrams was renormalized to cover the interval of exponents contained in the diagram.
Several bifurcation boundaries for orbits of low period as well as saddle-node and Hopf bifurcations have been recently reported theoretically and experimentally by Wieczorek et al. (2002) and Al-Hosiny et al. (2007). However, in addition to such bifurcation boundaries, our phase diagrams show details and regularities not observed before, now including all regions of periodic oscillations and the regions of chaos. Our phase diagrams show the inner details and structuring of stability domains, the regions where a recurring self-similar organization occurs and where it fails to exist.
In a pioneering work, Eriksson & Lindberg (2001, 2002) measured experimentally the location and magnitude of islands characterized by regular oscillations existing inside chaotic phases of semiconductor lasers. First, they identified a period-3 island by tuning the injection intensity for three fixed values of the frequency detuning. Then, by repeating measurements for finer detuning intervals, they cleverly managed to characterize a few islands of low period. Figure 1b corroborates their low-periodic islands and shows a myriad of additional islands of ever-increasing periods as discussed below. The figure also displays several novel features, in particular the existence of self-similarities of various kinds. Figure 1c displays an island with a familiar shrimp-shaped complex structure recorded when varying two parameters simultaneously (codimension-two phenomenon). Although well known in discrete-time dynamical systems (Gallas 1993, 1994, 1995), this peculiar shrimp-shaped structure was observed only quite recently in a continuous-time system, namely in CO2 lasers (Bonatto et al. 2005).
The above-mentioned striking and unexpected accumulation networks may be easily recognized from the phase diagrams of figure 2, which present successively magnified views of box B in figure 1b. Embedded in the chaotic region, there are regular and abundant networks of stable islands of periodic laser oscillations with unbounded periodicities. As figure 2a shows, the parameter networks living in the chaotic region bridge periodic laser oscillations of increasingly higher periodicities and which converge systematically towards well-defined and characteristic accumulation boundaries. As indicated schematically by the numbers in figure 2a, when moving along the dark central bodies of the islands one observes series of period-adding cascades of bifurcations, a characteristic signature of the experimentally elusive and rather challenging homoclinic route to chaos (Braun et al. 1992).
That stability organizes along specific directions in parameter space is a well-known fact for discrete-time dynamical systems (Gallas 1993, 1994, 1995). But that the same is true for continuous-time dynamical system is now made obvious by figure 2. But a rather unexpected novel feature clearly discernible in figure 2b is the surprising way in which individual period-adding bifurcation cascades are born. As seen, the pair of osculating white spines living inside the large dark island of period-10 oscillations splits as the period increases, leading to separate cascades that quickly give the impression of being uncorrelated owing to the strong compression experienced by the islands as the period increases more and more without bound. The white spine indicate loci of the most negative Lyapunov exponents and are loosely ‘equivalent’ to the familiar superstable orbits in discrete-time dynamical systems. The splitting process involves several specific metric properties, for instance the parameter separation of the islands accumulates to specific values while their volume decreases regularly with characteristic exponents (Bonatto & Gallas 2007).
As another remarkable result found in phase diagrams of differential equations, figure 3 illustrates two islands of regular oscillations with the same exquisite shape and structuring found very recently in a rather different scenario: in a discrete-time dynamical system with no critical points, i.e. 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. 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).
How can one detect experimentally the existence of accumulation networks embedded in chaos? One simple answer is provided by the bifurcation diagrams in figure 4, showing the unfolding of the period-adding cascades, as recorded independently for each dynamical variable, Ex, Ey and n, and the laser intensity . These diagrams clearly show the period-adding cascades, with the clear alternation of windows of chaos and periodic oscillations. In these diagrams, the numbers labelling periodic windows refer to the number of peaks present in one period of the respective variable. However, different variables display different number of peaks. Since the number of peaks is usually taken to label the ‘period’ of oscillation, one sees that such labels depend on the variable used to count the peaks. In particular, the numerical labels in figure 2 would be different had we used, say, the population inversion n instead of the laser amplitude I to count peaks. Clearly, the fact that local maxima of Ex and Ey occur here at distinct instants introduces a rather complicated Lissajous-like phase dependence responsible for the number of local maxima of I, which is very hard to predict. We observe that even in situations where direct measurement of the laser intensity might be difficult, a feasible alternative is to measure the spectrum of frequencies.
3. Lasers, electric circuits, Rössler and chemical oscillators
Are the regularities and accumulations described above for the optically injected semiconductor laser also found in other dynamical systems ruled by sets of nonlinear ordinary differential equations? The answer is yes. To corroborate our affirmative answer, figure 5 presents phase diagrams for several rather distinct dynamical systems without entering into discussions, due to space limitations. We simply give equations and parameters used to generate these phase diagrams. For additional details concerning the equations, we refer to the original references.
Interesting homoclinic phenomena are well known to exist in the following model of a CO2 laser with feedback, as described by Pisarchik et al. (2001):(3.1a)(3.1b)(3.1c)(3.1d)(3.1e)(3.1f)where f(x1)=Rx1/(1+αx1) is the feedback function. We used the same parameters considered by Pisarchik et al. (2001), namely Γ1=10.0643, Γ2=1.0643, α=32.8767, β=0.4286, k0=28.5714, k1=4.5556 and P0=0.016.
The third model considered here is a modification of the original Van der Pol circuit, proposed originally by Shinriki et al. (1981). It consists of an autonomous electronic circuit involving a resonant circuit and two nonlinear conductances, one positive and the other negative. Motivated by the interesting circuit of Shinriki et al. (1981), Freire et al. (1984) have shown shortly after that such circuit displays a rich variety of dynamical behaviours. The circuit is described by the following set of time-independent equations:(3.2a)(3.2b)(3.2c)where the bifurcation parameters shown in the figure are μ=G1+b1−a1 and δ=G2+b1, and we consider the same parameters used by Freire et al. (1984): C0=4.7 nF, C=100 nF, L=110 mH, a1/ωC=0.1, a3/ωC=6×10−4, b1/ωC=0.016 and b3/ωC=0.05.
The fourth model considered is the 40-year-old equations introduced by Rössler (1976)(3.3a)(3.3b)(3.3c)In our simulations, we fixed a≡0.2.
The next model considered represents a time-periodically forced chemical oscillator studied by Vance & Ross (1989) and Hou & Xin (1999) and described by the equations(3.4a)(3.4b)where(3.5)Here, following Vance & Ross (1989) and Hou & Xin (1999), we also considered their point P2=(j, Tc)=(0.8, 292) where the system exhibits self-oscillations with period T0=4.1627372 in absence of forcing. The other parameters are the following: ϵ=0.65; T*=885.843j+11.02Tc/(2.7j+11.02); η=T*/8827; D=8.2365×1010e−1/η/j; B=271.46/(ηT*); and β=1+4.08/j. As indicated in figure 5, the bifurcation parameters are A and τ. Our figures contain a wealth of new details, including cascades of accumulations.
Finally, we also considered Lorenz-84 low-order atmospheric circulation model (Lorenz 1984), a model containing rich accumulations and homoclinic phenomena with meteorological implications (Freire et al. 2007):(3.6a)(3.6b)(3.6c)As usual, we also considered the dynamics when fixing a=0.25 and b=4.
Each of these systems has a number of additional interesting aspects involving homoclinic phenomena and their consequences inside chaotic regions in parameter space. These are discussed in detail elsewhere. Here, the purpose is simply to emphasize that the accumulations and the accumulation of accumulations lying in ‘simple’ curves are generic phenomena found profusely in nonlinear systems of ordinary differential equations. Note that while differential equations have been around for quite a while, thus far no phase diagrams reporting the hierarchical structuring of the chaotic phases seem to exist in the literature.
4. Final remarks
We have shown that chaotic regions in phase diagrams of a number of dynamical systems ruled by sets of nonlinear ordinary differential equations contain very peculiar hierarchies of parameter networks ending in distinctive accumulation boundaries characterized by an infinite amount of accumulation of accumulations, with very rich and interesting structuring. The systems discussed here were semiconductor lasers with optical injection, CO2 lasers with feedback, autonomous electric circuits, the Rössler oscillator, Lorenz-84 low-order atmospheric circulation model and a chemical model. Since accumulation of accumulations was found in all these distinct physical models, we believe them to be the generic features of dynamical systems of codimension-two and higher.
The network of regularity islands found to exist embedded in chaotic phases may compromise applications depending on ‘smooth and clean domains of chaos’ such as secure communications. The parameter networks reported here pose an interesting question that remains to be investigated: in sharp contrast with discrete dynamical systems, where periods vary in discrete units, an appealing new possibility afforded by lasers is to study how periodicities defined by continuous real numbers evolve in phase diagrams when parameters are tuned. Additionally, another enticing question is whether or not it is possible to infer the presence of homoclinic orbits in phase space from regularities computed/measured solely in the parameter space. Are there clear parameter-space signatures of homoclinic orbits? Is it possible to recognize the location of parameter loci characterizing simple and multiple (degenerate) homoclinic phenomena in phase diagrams? These questions are the subject of a separate work.
The few points addressed here so far do not exhaust the richness of phenomena found in phase diagrams of continuous-time dynamical systems. As a last example, figure 6 illustrates families of isoperiodic parameter circles found in a CO2 laser with modulated losses. The equations and parameters used to produce figure 6 are those of Bonatto et al. (2005), with a single exception: here we use z0=0.18. When changing parameters inside such circles, the shape and the length of the periodic signals change but the number of peaks inside the circle period remains the same. An obvious question is that concerning the unfolding of these circles when other model parameters are varied, i.e. when considered as higher codimensional events. We have also seen a profusion of infinite hierarchies of nested spirals. This needs to be investigated.
As a last remark, it is perhaps interesting to emphasize that while accumulating regions of periodic orbits in optically driven lasers have been recently reported by Krauskopf & Wieczorek (2002), the phenomenon they describe is rather different from those reported here. For, as summarized in the comparisons presented in figure 7, they consider domains of periodic orbits, not regions of chaos as we do. Their accumulations involve saddle-node–Hopf bifurcations while ours do not. They investigate accumulations towards a fixed point while ours involve more complex objects. In addition, note that while Krauskopf & Wieczorek (2002) argue period-adding routes to chaos not to be present in semiconductor lasers, we find such routes to occur in profusion. A detailed discussion of these matters will be presented elsewhere.
The authors are indebted to Prof. F.T. Arecchi for drawing their attention to the Shilnikov phenomena in the CO2 laser model reported by Pisarchik et al. (2001). They also thank the Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil, for a doctoral fellowship (C.B.) and a senior research fellowship (J.A.C.G.). This work was supported by the Air Force Office of Scientific Research, through contract FA9550-07-1-0102.
One contribution of 14 to a Theme Issue ‘Experimental chaos II’.
- © 2007 The Royal Society