Analytically tractable reconstruction of singular hyperbolic and quasi-strange attractors of Lorenz-type systems

Abstract

We consider a certain three-dimensional piecewise linear system of Lorenz type in the cases of positive and negative saddle value, which is the sum of two eigenvalues of the saddle nearest to zero. This system was recently proposed and studied in \cite{belykh2019lorenz} for the case of the positive saddle value. Here we consider the main bifurcations leading to the birth of the strange attractor and provide its comparison with those of original smooth Lorenz model. For the case of a negative saddle value, we obtain an explicit forms of homoclinic and pitchfork bifurcations forming a codimension 1 cascade leading to the emergence of a quasi-strange attractor. This cascade reproduces a typical bifurcation route towards the birth of a quasi-strange attractor in smooth Lorenz-like systems with the negative saddle value. We support our analytical result with a numerical comparison with the Lorenz-Lyubimov-Zaks system.

Figures (8)

• Figure 1: Formation of a homoclinic orbit in in PWL system \ref{['sys:PWL_Lorenz']} with increasing $b$ (sketch from belykh2019lorenz).
• Figure 2: Comparison of bifurcation diagrams of PWL system \ref{['sys:PWL_Lorenz']} [the panel (a)] and the Lorenz system [panel (b)]. The curves $b_h$ and $l_h$ are homoclinic bifurcations, $b_{het}$ and $l_{het}$ are heteroclinic bifurcations, $b_{unq}$ and $l_{AH}$ are Andronov-Hopf bifurcations. In both systems, the main bifurcations occur in the same order. Lorenz strange attractor exists in yellow and green regions. The route points: [the panel (a)] a$(0.65, 1.5)$, b(0.65, 2), c$(0.65, 2.3)$, d$(0.65, 2.556)$, e$(0.65, 2.8)$, f$(0.65, 3.4)$; [the panel (b)] a(10, 10), b(13.927, 10), c(20, 10), d(24.06, 10), e(24.4, 10), f(28, 10). The parameters $\alpha = 2$, $\nu = 0.65$, $\lambda = 0.294$, $\omega = 2$, and $\delta = 0.588$ of PWL system \ref{['sys:PWL_Lorenz']} are fixed. The line $b_{cr}=3.95548$ corresponds to destruction of the attractor via appearance of sliding motions. Phase portraits for corresponding route's points are in Fig. \ref{['fig:phase_port_positive']}.
• Figure 3: Comparison of phase portraits of PWL system \ref{['sys:PWL_Lorenz']} and the Lorenz system when the bifurcation parameter changes along the route $\text{a}\rightarrow \text{f}$ (see diagrams in Fig. \ref{['fig:bif_diag_positive']}). The rows (a)--(f) correspond to points $\text{a} - \text{f}$ in Fig. \ref{['fig:bif_diag_positive']} with the same parameter values.
• Figure 4: The first bifurcations of the factor map \ref{['map:master']}. (a) $\gamma$ = 0.7. Fixed points $e_{l,r}$ attract the entire interval. (b) $\gamma$ = 0.8. Attractors are fixed points $p_{l,r}$, born from $e_{l,r}$ as a result of a transcritical bifurcation. (c) $\gamma$ = 0.9. The points $p_{l,r}$ formed a superstable fixed point at zero. (d) $\gamma$ = 1. The first stable orbit $p^2$ of period 2 was born. (e) $\gamma$ = 1.277.Attractors are orbits $p_{1,2}^2$ of period two, born from $p^2$ as a result of a pitchfork bifurcation. (f) $\gamma$ = 1.325. Orbits $p_{1,2}^2$ stuck with zero, thus forming a superstable orbit $p^4$ of period 4.
• Figure 5: Comparison of bifurcation routes $\text{a}-\text{f}$ for $\nu>1$. (a) Bifurcation diagram of PWL system \ref{['sys:PWL_Lorenz']}. The saddle index is fixed $\nu=1.25$. The color shows the logarithm of the factor map \ref{['map:master']} attractor multiplier $\ln\mu$. The dashed area corresponds to coexistence of stable fixed points $e_{l,r}$ and the strange attractor. The points: $\text{a}(1.25, 1.4)$, $\text{b}(1.25,1.8)$, $\text{c}(1.25,2)$, $\text{d}(1.25,2.3)$, $\text{e}(1.25,2.6)$, $\text{f}(1.25,2.65)$, $\text{g}(1.25,3.8)$. Other parameters are the same as in Fig. \ref{['fig:bif_diag_positive']}. (b) LLZ system \ref{['sys:Lyubimov']} (a sketch from kazakov2021bifurcations). Orange stands for maximal positive Lyapunov exponent and blue for zero. The route is defined by $r\approx\frac{0.881}{D}$. The points: $\text{a}(0.0938, 9.38)$, $\text{b}(0.0915,9.64)$, $\text{c}(0.08,11.01)$, $\text{d}(0.0711,12.41)$, $\text{e}(0.0624,14.2)$, $\text{f}(0.0586,15.1)$ , $\text{g}(0.05,17.8)$. The parameters: $\sigma=10$, $\beta=\frac{8}{3}$. For both systems The routes intersect the same bifurcations: $b_{AH}$ and $l_{AH}$ - Andronov-Hopf bifurcation, $H_1$ - the first (one-round) homoclinic butterfly, $F_2$ - pitchfork bifurcation, $H_2$ - two-round homoclinic butterfly, $H_\infty$ - limiting curve of quasi-strange Lorenz-type attractor birth. In both systems, there are periodic windows in the region above the curve $H_{\infty}$. The routes points are the same as in Fig. \ref{['fig:map_v>1']}. The point $\text{g}$ is the quasi-strange attractor. The corresponding phase portraits see in Fig. \ref{['fig:PWL_Lubimov_compare']} and Fig. \ref{['fig:Atractor_Lorenz_like']} for $\text{g}$.