Diverging period and vanishing dissipation: Families of periodic sinks in the quasi-conservative case

Figure 1.  "Dribbling" a possible singularity at $|b| = 1$, in 4-dimensional space $(x,y,b,a)$ to be able to then approach it from elsewhere. Starting at step 1 on a period-7 saddle in the conservative case, follow branches of continuation curves in $(x,y,a,b)$, by switching independent variable at each turning point 2, 3, 4. At steps 3 and 4 one can follow a bifurcation curve with known continuation methods. Upper and lower curves are respectively period-doubling and saddle-node bifurcation curves

Figure 2.  The two conservative cases to initialize a dribbling method. Left: for $b = -1$ and $a = 0.244$ big dots represent a period 14 saddle. Right: for $b = 1$ and $a = -0.719$ big dots represent a period 7 saddle

Figure 3.  Adjustments in renormalization scheme of the saddle node bifurcation value. If Newton's method initialized using linear prediction seems not to converge, adjust initial guess only in one variable $y$, and try again. This figure illustrates dependence of the final error in the calculation of $a_{sn}$ from $y-$adjusted guess

Figure 4.  Blow-up of same error function of Fig. 3 but with $y-$range $2\cdot 10^{-4}$. The minimal error can now be detected using an automatic algorithm in the suitable interval where the error function is observed to be smooth

Figure 5.  Stability ranges in the $a$ parameter for $b = 1-10^{-5}$ and periodic orbits of period $p$ between 6 and 50. For $a\equiv{\bar a}_{max}(\varepsilon) = -0.93294$ there are a maximum of 10 coexisting attractors of period between $p = 11$ and $p_1 = 20$

Figure 6.  Width of the stability ranges in the $a$ parameter for $b = 1-10^{-5}$ and periodic orbits of period $p$ between 6 and 60. The rate of fast exponential convergence starts around $p = p_{2}\equiv 25$

Figure 7.  Stability ranges in the $a-a_{min}$ parameter for $b = 1-10^{-5}$ and periodic orbits of period $p$ between 34 and 70. The values of $a_{min}$ is the minimum value of $a_{sn}^{(p)}$ (reached for $p = p_{3} = 38$). Observe that $p = 36$ is the largest period such that $a_{sn}^{(p)}\leq a_{sn}^{(36)}$ for any $p\geq 36$

Figure 8.  Width of the stability ranges in the $a$ parameter and coexistence in $p$ of periodic attractors with $\varepsilon = 10^{-6}, 10^{-7}$ and $10^{-8}$. Left: $b = -1+\varepsilon$. Right: $b = 1-\varepsilon$

Figure 9.  Rate of convergence in $p$ of the stability range $a_{pd}(\varepsilon) - a_{sn}(\varepsilon)$ with $\varepsilon = 10^{-5}, 10^{-6}, 10^{-7}$ and $10^{-8}$. Left: $b = -1+\varepsilon$ Right: $b = 1-\varepsilon$. The negative value of the slope $m_\varepsilon$ in logarithmic scale increases as $\varepsilon$ decreases.

Figure 10.  Stability ranges in the $a-a_{min}$ parameter with $\varepsilon = 10^{-6}, 10^{-7}$ and $10^{-8}$. Left: $b = -1+\varepsilon$. Right: $b = 1-\varepsilon$

Figure 11.  Coexistence of periodic attractors (isolated points), all other points are orbits of a discretized segment: on the unstable manifold (and inside a $10^{-8}$ neighborhood) of the fixed point (rightmost big dot). $b = -1+\varepsilon$. Up: ${\bar a}_{max}(10^{-5}) = 0.0769$. Down: for ${\bar a}_{max}(10^{-6}) = 0.0549$ the scale is five times smaller

Figure 12.  Stability range in $(a,\nu)$, $b = 0.5$, for two periodic orbits with rotation number $\omega = 3/5$ and $\omega = 8/13$. The border curves corresponds to values of $a_{sn}(\nu)$ for which the residue is $0$. The middle curve corresponds, given $b$ and $\nu$, to values of $a$ for which the residue is maximum