Border-collision period-doubling scenario

Using a one-dimensional dynamical system, representing a Poincaré return map for dynamical systems of the Lorenz type, we investigate the border-collision period-doubling bifurcation scenario. In contrast to the classical period-doubling scenario, this scenario is formed by a sequence of pairs of bifurcations, whereby each pair consists of a border-collision bifurcation and a pitchfork bifurcation. The characteristic properties of this scenario, like symmetry-breaking and symmetry-recovering as well as emergence of coexisting attractors and noninvariant attractive sets, are investigated.

Figure 1

Typical shapes of the system function $f(x,\alpha )$ for different values of the parameter $\alpha$.

Figure 2

Border-collision period-doubling scenario. Shown are the periods $T$ (logarithmic plot) and the Lyapunov exponents $\lambda$. Note, that these diagrams are identical with the corresponding diagrams of the period-doubling scenario in the case of the logistic map.

Figure 3

Border-collision period-doubling scenario. This bifurcation diagram shows remarkable differences compared with that of the classical period-doubling scenario.

Figure 4

First border-collision bifurcation at $\alpha ={\alpha }_1^{\mathit{bc}}=2$. Shown are the system function (thick line) and its second iterated function (thin line) before the bifurcation (a) and after the bifurcation (b). (c) is a blowup of the rectangle marked in (b). The dotted lines in (b) and (c) mark the functions $f_l$ and $f_r$ outside their domains.

Figure 5

Analytical results about the attractors of system (1). The following bifurcation points are marked: ${\alpha }^t$ transcritical bifurcation; ${\alpha }_1^{\mathit{bc}}$ first border collision bifurcation; ${\alpha }_1^p$, first pitchfork bifurcation; ${\alpha }_2^{\mathit{bc}}$ second border collision bifurcation; and ${\alpha }_2^p$ second pitchfork bifurcation. The points $x_1^{*}$, $x_2^{*}$, and $x_3^{*}$ are the fixed points. The points $x_4^{*}$ and $x_5^{*}$ are fixed points between ${\alpha }^t$ and ${\alpha }_1^{\mathit{bc}}$ and build a limit cycle with period two after ${\alpha }_1^{\mathit{bc}}$. The points $x_1^{**}$, $x_2^{**}$, $x_3^{**}$, and $x_4^{**}$ build two coexisting limit cycles with period two between ${\alpha }_1^p$ and ${\alpha }_2^{\mathit{bc}}$ and limit cycle with period four after ${\alpha }_2^{\mathit{bc}}$.

Figure 6

Third border-collision bifurcation at $\alpha ={\alpha }_3^{\mathit{bc}}\approx 3.4985$. Two coexisting asymmetric limit cycles with period $T=4$ before the bifurcation (a), (b). A symmetric limit cycle with period $T=8$ after the bifurcation (c).

Figure 7

Schematic representation of the classical period-doubling scenario (a) and the border-collision period-doubling scenario (b). The dashed boxes mark the regions that can be denoted as one step of the corresponding scenario.

Figure 8

Mean points $\overline{x}$ of the attractors within the border-collision period-doubling scenario obtained for two symmetric initial values [$x\left(0\right)=0.25$ and $x\left(0\right)=0.75$]. (b) and (c) are blowups of the regions marked in (a) and (b).

Figure 9

Band-merging cascade in system (1). See text for detailed description.

Figure 10

Boundaries of chaotic attractors. (a) are the six functions defining the boundaries of three-band attractors. (b) are the eight functions (thick lines) that define together with the six functions of (a) (thin lines) the 14 boundaries of seven-band attractors. Note that these function also reveal the basic structure of the bifurcation diagram, including periodic windows presented in Figs. 3, 9.

Figure 11

Border-collision period-doubling scenario within the three-periodic window (a). Enlarged are the upper part (b), the middle part (c), and the lower part (d) of the scenario.

Figure 12

Behavior of system (1) at the point $\alpha ={\alpha }_2^{\mathit{bc}}=1+\sqrt{5}$ of the second border-collision bifurcation. Upper part: critical points $A$, $B$, $C$ and the mapping of their neighborhoods onto each other. Lower part: three steps of this mapping for the right-side open neighborhood ${\mathcal{U}}^{+}\left(A\right)$.

Figure 13

Second border-collision bifurcation at $\alpha ={\alpha }_2^{\mathit{bc}}=1+\sqrt{5}$. Numerical observed behavior of system (1) before the bifurcation (a), at the bifurcation point (b), and after the bifurcation (c).