Basin size evolution between dissipative and conservative limits

Recent methods for stabilizing systems like, e.g., loss-modulated ${\mathrm{CO}}_2$ lasers, involve inducing controlled monostability via slow parameter modulations. However, such stabilization methods presuppose detailed knowledge of the structure and size of basins of attraction. In this Brief Report, we numerically investigate basin size evolution when parameters are varied between dissipative and conservative limits. Basin volumes shrink fast as the conservative limit is approached, being well approximated by Gaussian profiles, independently of the period. Basin shrinkage and vanishing is due to the absence of bounded motions in the Hamiltonian limit. In addition, we find basin volume to remain essentially constant along a peculiar parameter path along which it is possible to recover the dissipation rate solely from metric properties of self-similar structures in phase-space.

Figure 1

(Color online) Top: Schematic view of boundaries between stability regions. Numbers label bifurcation loci: 1 marks the $0\to 1$ saddle-node bifurcation, 2 marks the $1\to 2$ bifurcation while 4 marks the $2\to 4$ bifurcation. $U$ and $L$ refer to the upper and lower branches of the eigenvalue path (see text). The dashed rectangular box is shown magnified in the lower part of the figure. Bottom: Phase diagram showing the intricate alternation of stability domains underlying $L$ branch of the eigenvalue path $E(a,b)$. Different shadings denote different periods. Encircled numbers indicate the periodicity of the underlying domain. White represents chaos. The basin of unbounded attractors is indicated by $-\infty$.

Figure 2

(Color online) Illustrative basins computed for four different set of parameters along the $2\to 4$ bifurcation line, defined by Eq. (3): (a) $(a,b)=(0.9125,0.9)$; (b) (1.042, 0.16); (c) (2.05, $-0.4$); (d) (3.25, $-0.8$). The gray shading indicates the basin of $-\infty$, black gives the basin of the periodic motion.

Figure 3

(a) Basin volume for the three lowest periods of the $1\times 2^n$ cascade, as a function of the dissipation $b$. (b) Overlap of five curves, showing basin volume as a function of the dissipation $b$ for periods 8, 16, 32, 64, 128, and 256. The five curves overlap almost identically. The only noticeable differences involve intervals where multistability is present, e.g., the period-3 window indicated by 3.

Figure 4

The standard deviation of Gaussian fits to basin volume as a function of basin periodicity. See text. The rightmost point represents a rough estimate at the end of the $1\times 2^n$ bifurcation cascade.