Homoclinic chaos and its organization in a nonlinear optics model

We developed a powerful computational approach to elaborate on the onset mechanisms of deterministic chaos due to complex homoclinic bifurcations in diverse systems. Its core is the reduction of phase space dynamics to symbolic binary representations that lets one detect regions of simple and complex dynamics as well as fine organization structures of the latter in parameter space. Massively parallel simulations shorten the computational time to disclose highly detailed bifurcation diagrams to a few seconds.

Figure 1

(a) $(\beta ,-D_{23})$ phase space projection showing the primary homoclinic orbit (red, coded as $\left\{1\right\}$) splitting leftward/rightward (green/blue, $\{10...\}$ or $\{11...\}$) when the separatrix ${\mathrm{\Gamma }}_1$ misses the saddle $O$ (black dot) after completing a single turn around the saddle focus $C^{+}$, with the Lorenz attractor (in gray) in the background. (b) Chaotic transient of ${\mathrm{\Gamma }}_1$ generating a binary sequence starting with $\{10100101...\}$. (c) Time evolutions of the $\beta$ coordinate of ${\mathrm{\Gamma }}_1$ [in (b)] and of a close trajectory (red), and their binary codes, before they diverge. (d) Two stable symmetric POs coded as $\left\{\overline{01}\right\}$ and $\left\{\overline{0011}\right\}$. (e) Heteroclinic connections (red $\left\{10\overline{1}\right\}$, blue symmetric counterpart) at the $T_1$ point (Fig. 2). (f) Samples ($P_j$) of the primary homoclinic orbit morphing to a double loop after the inclination flip ${\text{IF}}_1$ on the curve $H_0$ in the $(a,b)$-parameter plane in Fig. 2; here, $\sigma =1.5$.

Figure 2

$(a,b)$-parameter sweep of [5–12] length reveals an abundance of homoclinic bifurcations emerging from two cod-2 points, ${\text{IF}}_1$ and ${\text{IF}}_2$, on $H_0$, that correspond to the primary homoclinic butterfly of saddle $O$, along with self-similar characteristic spirals around the $T$ points, labeled $T_{0,1,2}$, corresponding to distinct heteroclinic cycles between $O$ and saddle foci $C^{\pm }$. Cod-2 Bogdanov-Takens (BT) unfolding includes Andronov-Hopf ${\text{AH}}_0,{\text{AH}}_{1,2}$ and pitchfork PF bifurcation curves for $O$ and $C^{\pm }$, respectively; here, $\sigma =1.5$.

Figure 3

(a) Short [8–15] and (b), (d) long [100–123] length $(a,b)$ sweeps reveal fine self-similar organization of homo- and heteroclinic bifurcations underlying the regions of chaotic and regular dynamics of model (1) for $\sigma =1.5$. A small area (white box) in (a) is magnified with a longer [15–22] sweep in (c). (b) and (d) reveal stability windows (solid colors) within “noisy” regions of structurally unstable chaos; white lines demarcate boundaries of some stability windows.

Figure 4

[2–9] length sweep discloses organization of homo/heteroclinic bifurcations originating from cod-2 inclination flip ${\text{IF}}_2$ and multiple $T$ points: Primary $T_0$ coded as $\left\{1\overline{0}\right\}$, secondary $T_1$ as $\left\{10\overline{1}\right\}$, and a pair $T_2^1-T_2^2$ with code $\left\{11\overline{0}\right\}$ separated by a saddle ($S$) in the $(a,b)$-parameter plane; here, $\sigma =10$. Inset (a) shows a larger $(a,b)$ sweep of [1–7] length; (b) [16–23]-long sweep depicts dense loci of homoclinic bifurcation curves originating from ${\text{IF}}_2$.

Figure 5

Fragment of the 3D $(a,b,\sigma )$-parameter space (based on a $2000\times 2000\times 2000$ grid) depicting various nested elliptic and hyperbolic paraboloids whose contour curves appear as spirals around $T$ points, and/or concentric circles near saddles in the biparametric sweeps in Figs. 2, 3, 4.

Figure 6

Long [1000–1999]-length sweeps to detect a multiplicity of stability windows [solid colors; dark red due to stable PO $\left\{\overline{0011}\right\}$ in Fig. 1] within noisy/multi-color regions of chaos adjacent to $I{F}_1$ and $I{F}_2$ points in the $(a,b)$-parameter plane using PC algorithm in (a) and (c), and LZ-complexity in (b) and (d). Sweeps at $\sigma =1.5$ (a,b) and $\sigma =10$ (c,d) to compare with Fig. 2 and 4.