Hierarchical collapse of regular islands via dissipation

The effect of dissipation on tori can be nicely described using the dissipative twist map

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:map3} x_{n+1} = J\, x_n\cos{(\psi)} - y_n \sin{(\psi)}, \nonumber \end{align} \tag{ 2 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:map4} y_{n+1} &= J\, x_n\sin{(\psi)} + y_n \cos{(\psi)},\nonumber \\ \nonumber \end{align} \tag{ 3 }$$

where ($x_n, y_n$) may represent canonical variables. The Jacobian of the above map is equal to J. Therefore, for J  =  1 the above map is the conservative twist map [1]. Here $\psi=2\pi\, \alpha_0$ with $\alpha_0$ being the rotation number. For J  =  1 the dynamics obtained from (2) and (3) is a rational torus when $\alpha_0$ assumes a rational value k/q, with k and q being integers. In such cases a rational torus with period-q is obtained. For irrational values of $\alpha_0$, the dynamics is an irrational torus. The map (2) and (3) allows us to understand the basic general effect of our dissipative model acting on rational/irrational tori from the conservative case. Similar dissipative effects on the rational and irrational tori are expected for the dynamics of the map (1).

The explicit time evolution of the variables ($x_n, y_n$) can be obtained by applying the discrete Laplace transform [30] in equations (2) and (3)

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\zt}[2]{\tilde{#1}(#2)} \displaystyle \label{eq:zmap3} \zt{x}{z} =&~ \frac{(z-a) z}{z^2 + a^2 J + b^2 J - a z (J + 1)} x_0 \nonumber \\ \nonumber &-\frac{bz}{z^2 + a^2 J + b^2 J - a z (J + 1)} y_0, \nonumber \end{align} \tag{ 4 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\zt}[2]{\tilde{#1}(#2)} \displaystyle \label{eq:zmap4} \zt{y}{z} =&~ \frac{bz J}{z^2 + a^2 J + b^2 J - a z(J + 1)} x_0 \nonumber \\ \nonumber & +\frac{(z - a J) z}{z^2 + a^2 J + b^2 J - a z (J + 1)} y_0. \nonumber \end{align} \tag{ 5 }$$

After long but straightforward calculations which involve partial fractions decomposition of the coefficients for the inverse Z-transform evaluation, we obtain:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \nonumber x_n =&~ J^{n/2} \left[ \cos{(n\theta)} + \frac{a(J - 1)}{\sqrt{4b^2 J - a^2 (J - 1)^2}} \sin{(n\theta)} \right] x_0 \nonumber \\ \label{eq:x-solve} & -\left[ \frac{2b J^{n/2}}{\sqrt{4b^2 J - a^2 (J - 1)^2}} \sin{(n\theta)} \right] y_0, \nonumber \\ \nonumber y_n =&~ \left[ \frac{2b J^{\frac{n+2}{2}}}{\sqrt{4b^2 J - a^2 (J - 1)^2}} \sin{(n\theta)} \right] x_0 \nonumber \\ & +J^{n/2} \left[ \cos{(n\theta)} - \frac{a(J - 1)}{\sqrt{4b^2 J - a^2 (J - 1)^2}} \sin{(n\theta)} \right] y_0. \nonumber \end{align} \tag{ 6 }$$

The angle θ is defined through:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\eq}[1]{equation~(\ref{#1})} \displaystyle \label{eq:y-solve} \cos{(\theta)} \equiv \frac{a (J + 1)}{2 \sqrt{J}}, \nonumber \end{align} \tag{ 7 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\eq}[1]{equation~(\ref{#1})} \displaystyle \sin{(\theta)} \equiv \frac{\sqrt{4 b^2 J - a^2 (J - 1)^2} }{2 \sqrt{J}}, \nonumber \end{align} \tag{ 8 }$$

where we introduce factor 1/(2J^1/2) such that $\sin^2{(\theta)} + \cos^2{(\theta)} = 1$. Thus,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \theta = \tan^{-1}\left[ {\frac{\sqrt{4 b^2 J - a^2 (J - 1)^2} }{a (J + 1)} } \right], \nonumber \end{align} \tag{ 9 }$$

being $a = \cos{(\psi)}$ and $b = \sin{(\psi)}$. For one numerical example we use $\psi = \pi^2 / 2$, $(x_0, y_0) = (0.1, 0.21)$ and J  =  0.8 to plot in figure 1 the iterations of maps (2) and (3) together with analytical solutions given by equations (6) and (7).

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The analytical solutions show that dissipative effects destroy the tori, and they collapse from ($x_n, y_n$) to (0,0) with a time decay proportional to $J^{n\sin{(\psi)}}={\rm e}^{n\sin{(\psi)}\ln{(J)}}$. For 0  <  J  <  1, which is the case of dissipation, $\ln{(J)}<1$ and the decay becomes proportional to ${\rm e}^{-n\sin{(\psi)}\ln{(J)}}$. Thus, the dissipative model used here induces an exponential convergence in time. However, for tiny dissipation this decay can be very slow, in fact linear. This can be checked by expanding the exponential in a Taylor series around $\ln{(J)}\approx 0$. As an example, using J  =  0.999, after n  =  1000 iterations we have $x_n=y_n=0.3677$, and for J  =  0.99 we have $x_n=y_n=4\times10^{-5}$ for the same number of iterations. Results from this section are applied for rational and irrational tori.

In area preserving mappings, by slightly perturbing a two-dimensional integrable system, it is possible to obtain a perturbed twist mapping [1]. By conveniently linearizing such a map around a fixed point the result is the standard map (1) with K essentially proportional to $\newcommand{\e}{{\rm e}} \epsilon k_0^2V_0$ [5] and J  =  1. Here $\newcommand{\e}{{\rm e}} \epsilon\ll 1$ is the perturbation parameter, V₀ is the perturbation calculated at the resonance condition and k₀ is the order of the nonlinear resonance. This means that any attempt to understand the effect of dissipation in the presence of nonlinear resonances can use map (1) with $J\ne 1$ and $K\ll1$. To check this, figure 2(a) demonstrates the convergency of the interpolated radius $r_n=\sqrt{x_n^2+p_n^2}$ for one initial condition and K  =  0.1, for which the map (1) has only one fixed point at $x_1=1/2, p_1=0$. Black for J  =  0.9999, red for J  =  0.9995, green for J  =  0.999, blue for J  =  0.995 and yellow for J  =  0.99. In these cases the exponential decay ${\rm e}^{-\beta n}$ has β given respectively by $2.3\times 10^{-4}$, $1.1\times 10^{-3}$ $2.2\times 10^{-3}$, $1.1\times 10^{-2}$ and $2.2\times 10^{-2}$. In figure 2(b) we use K  =  2.5 (same values of J) to describe the results regarding higher order resonances discussed in section 4.4. In these cases β is given respectively by $4.7\times 10^{-4}$, $2.4\times 10^{-3}$, $4.8\times 10^{-3}$, $2.4\times 10^{-2}$ and $4.8\times 10^{-2}$.

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We start this section discussing the general behaviour of stable and unstable periodic orbits when arbitrary small dissipation is considered. Figure 3(a) shows the phase-space for the conservative case ($\gamma=0$). The two per-1 stable points, also known as primary resonances, are localized at p  =  0 (see the red line in figure 3(a)) with $x=\pm1/2$. The four stable elliptic points, or secondary resonances, related to the per-4 orbit are located at p  =  0, $x\sim\pm 0.32$ and $p\sim\pm 0.36$, $x\sim\pm 0.32$, which can be seen figure 3(b), which is an amplification of the domain delimited by a red box drawn in the upper-left part of figure 3(a). These secondary resonances are surrounded by their corresponding rational/irrational tori and also by some smaller elliptic points, the ternary resonances and so on. Since the set of stable points related to the primary and secondary resonances will be discussed in more detail later, we introduce the notation $P_1^{\pm}$ and $P_4^{\pm}$, respectively. The sign refers to the sign of x, or the location of the stable points, i.e. left for negative and right for positive x values. Moreover, as some specific periodic orbits will be further investigated in the section 4.3, we also define the notation $P_1^{\pm (\tau)}$ and $P_4^{\pm (\tau)}$, where the sign and subscript-index have the same meaning introduced in the last sentence, while the superscript-index τ is the period of the orbit which will be discussed later. For example, $P_4^{+(28)}$ refers to the point of a periodic orbit with period 28, which is located around the main island with period-4. The remaining phase-space dynamics is chaotic, beside some tiny stable points, not visible in the resolution shown.

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The domains with regular motion are not stable under dissipative perturbation. By adding an arbitrary small amount of dissipation to the dynamics, $\gamma=10^{-3}$ for instance, we observe in figure 3(c) that rational/irrational tori are destroyed and the stable points become attracting centres or sinks [21, 22]. In other words, we can say the tori lose their energy and converge to the stable elliptic point they belong to. This is a very typical behaviour in the conservative to dissipative transition, also shown analytically in section 3. While elliptic points are transformed into sinks and surrounding orbits converge to it, chaotic orbits usually converge to the chaotic attractor. In other words, we can say that the phase-space near to the conservative limit is dominated by periodic attractors with different periods [25–27], where stable points or resonances become attractors and tori are destroyed giving rise to the basin of attraction. Roughly speaking, when slowly increasing the amount of dissipation in the system, points on the tori become basins of attraction for the periodic points and the chaotic motion is replaced by a very long chaotic transient motion that occurs before the trajectory converges to the chaotic attractor. The asymptotic state of system is extremely susceptible to transient behaviour.

In fact, even though the above behaviour seems to be simple, it is not. We have to remember that this is the origin of the basin of attraction in dissipative systems which has complicated and fractal structures [15]. The border line between the basin of attraction for distinct attractors is an invariant line. For small dissipation the stable irrational tori surrounding the elliptic points apparently just converge to their corresponding elliptic point. However, what happens to higher-order resonances which live at the border between irrational tori and the chaotic motion? We remember that each higher-order resonance has its own surrounding irrational tori. We can also ask what decides if a chaotic trajectory from the conservative case converges to one specific elliptic point, or to a fractal attractor as dissipation increases? These are some of the important questions that we discuss in this work.

We start analyzing how one irrational torus is deformed by different dissipation intensities. For this we plot figure 4, which is a magnification of the box drawn in figure 3(a), for different values of γ. Figure 4(a) presents the conservative case ($\gamma=0$) and the red curve is the KAM torus generated by the initial condition $(x_0, p_0)=(-0.31, 0.00)$, chosen in the neighborhood of a stable point localized at $P_4^-=(-0.32, 0.00)$. Setting $\gamma=10^{-4}$, the main visible effect is that the torus becomes thick, as seen in figure 4(b). In this case, the dissipation is not strong enough to allow the perturbed torus to break and converge exactly to the nearest stable point labeled $P_4^-$. Although the trajectory is closer to $P_4^-$ when compared to the torus from the conservative case. Increasing dissipation to $\gamma=10^{-1}$ the orbit is completely broken and converges to a point apart from $P_4^-$, as shown in figure 4(c). Increasing the dissipation more (figure 4(d)) makes the convergence become faster and the trajectory is attracted by the sink (attractor) at $(x, p)=(-0.5, 0.0)$, which is the point $P_1^-$ from the conservative case.

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Another way to see the importance of dissipation is by looking at figure 5, where in (a) and (b) we display the phase space dynamics for the same values of γ used in figures 4(c) and (d), namely $\gamma=10^{-1}$ and $\gamma=0.3$, respectively. In figure 5 (a) there are six regions with a higher-density of points when compared to the rest of the phase-space. These regions are close to the stable points $P_1^{\pm}$ and $P_4^{\pm}$ from figure 3(a), which are now transformed into attractors. Increasing the values of the dissipation, figure 5 (b) shows that the higher-density of points are now close to $P_1^{\pm}$, a consequence of the fact that $P_4^{\pm}$ became unstable. Consequently, all attractors generated with higher-order resonances converge, due to dissipation, to attractors associated to primary resonances. Figures 5(a) and (b) can be compared to figure 3(c), where the dissipation is smaller and the density points increase also around higher-order resonances, the behaviour of which is explained next.

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Figure 5(c) displays the final orbital location (x_f) as a function of γ. To obtain this picture we use two ICs, one at $(x_0, p_0)=(-0.31, 0.00)$ (black curve), localized near to the stable point $P_4^-$, and the other one (red curve) at $(x_0, p_0)=(-0.3630, -0.0458)$, related to the per-$7\times4$ from figure 3(b). With these ICs the standard map was iterated 10⁴ times for 200 values of the dissipation parameter in the range $\gamma=[0:1]$. For each value of γ distinct x_f are reached. Close to $\gamma\sim 0.03$ the per-$7\times4$ around the island from figure 3(b) collapse together with the black curve, which is related to the IC from the stable point $P_4^-$. Thus, for $\gamma\gtrsim 0.03$ the per-$7\times4$ disappears. For $\gamma\gtrsim 0.2$, both trajectories converge to the point related to $P_4^-$.

Summarizing this part, we conclude that when a small amount of dissipation is introduced in conservative systems, tori around an elliptic point are destroyed, becoming points which belong to the basin of attraction of this elliptic point, which is now an attractor. For tiny dissipation the number of attractors should be equal to the number of resonances from the conservative case, which can be huge. As dissipation increases, attractors related to higher-order resonances are destroyed first, leading to a hierarchy of the destruction of attractors. For strong dissipation, only primary resonances tend to survive. With these results we are tempted to conclude that the hierarchy of convergences, as dissipation increases, follows the simple rule: higher-order resonances converge to lower order resonances and so on. However, as we will see next, the hierarchy follows a more complicated scheme.

When adding weak dissipation, it is very difficult to know in general to which attractor a given IC belongs to. Besides the ICs on the torus close to the elliptic points, all other points in the phase space can belong to a distinct basin of attractions. In this section we study this in more detail. In figure 6 we plot the initial position versus convergence point ($x_0 \times x_f$) after n  =  10⁴ iterations, considering the interval x₀  =  [−0.5:0.5], and four distinct values of dissipation. Starting with figure 6(a), where $\gamma=10^{-5}$, we can clearly identify the ICs chosen in the neighborhood of some stable point (from the conservative regime) converging to itself. For example, the IC $(x_0, p_0)=(-0.30, 0.00)$, which is inside the regular island close to $P_4^-$, generates a trajectory that converges to x_f  =  −0.32, as indicated by the blue-dashed line. A similar behaviour occurs for other ICs, chosen close to a lower resonance, as can be seen for x_f  =  −0.50, and indicated by the red-dashed line. This is the location of the largest regular island where the stable point $P_1^-$ exists. Otherwise, initial values of x₀ chosen outside the regular islands are related to the chaotic asymptotic behaviour. Such behaviours become much more evident as dissipation increases, as can be seen in figures 6(b)–(d). For $\gamma\simeq 0.26$, we see in figure 6(c) that ICs inside the regular islands around $P_4^\pm$ converge to $x_f=\pm 0.32$, while ICs started inside the islands related to stable points $P_1^\pm$, converge to $x_f=\pm 0.50$. However, when dissipation increases to $\gamma=0.27$, we clearly see in figure 6(d) that all ICs chosen inside any regular island converge to the same final point x_f  =  +0.50 or  −0.50.

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The dependence of x_f over the choice of initial position x₀ can be nicely explored using a mixed-plot, which was proposed recently to describe generalized bifurcation diagrams [11]. Figure 7 displays the mixed-plot, where red, yellow to white colours indicate increasing positive values of x_f, and cyan, blue to black colours belong to increasing negative values of x_f. ICs are taken along the line p₀  =  0 and x₀  =  [−0.5, 0.5]. Near the conservative limit ($\gamma\rightarrow 0$), the rational/irrational tori and the chaotic trajectory surrounding the stable points $P_1^{\pm}$ and $P_4^{\pm}$ converge respectively to these points. See for instance, that the rational/irrational tori surrounding $P_1^{\pm}$ converge to $x_f=\pm 0.50$, which are the white (represented by sign  +) and black (represented by  −) points. Rational/irrational tori surrounding $P_4^{\pm}$, and ICs inside the chaotic domain, converge to $x_f\sim\pm 0.32$, and are related to yellow for  +  and blue for  −  convergence points. Moreover, the ICs chosen in the chaotic domain, between the regions associated to $P_4^-$ and $P_4^+$, converge to other points in the phase-space, probably to the chaotic attractor. This can be recognized by the mixture of colours designed for x_f in figure 7(a). As dissipation increases, we see the evolution of the final points x_f. For $\gamma \gtrsim 0.18$, the stable points $P_4^{\pm}$ do not attract the surrounding tori anymore (yellow and blue points around these points disappear). The corresponding ICs now converge to the points $P_1^{\pm}$ represented by white and black colours. The large variety of colours for x_f also start to disappear, but in a complicated and intricate way. For stronger values of dissipation all the ICs converge to the points $P_1^{\pm}$, which are the points remaining from the primary resonances in the conservative limit. Figure 7(b) illustrates very well the complex scenario we found. It is a magnification between chaotic and regular ICs, which is a portion of figure 3(a) between the points $P_1^+$ and $P_4^+$. The larger yellow stripe is reminiscent of ICs from the conservative case which were close to the border of the last tori around $P_1^+$ [a similar (blue) stripe exists for $P_1^-$]. Please observe that the yellow stripe starts for $\gamma=0$ at the border of the large yellow stripe from figure 3(a). All these ICs start close to $P_1^-$.

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The third strategy used to study the hierarchical destruction of regular islands is based on the application of the largest finite-time Lyapunov exponents (FTLEs). It will help us to check if there is a stability change when ICs change their attractor due to small dissipation. Since our interest is related to the study of the convergence properties of stable orbits under dissipative effects, the largest FTLEs, represented by λ, has negative values and becomes zero at the bifurcation points. For more details about the application of this method to study conservative systems we refer the readers to see [11]. From the numerical point of view we calculate the FTLEs using Benettin's algorithm following [31] and [32], which includes the Gram–Schmidt re-orthonormalization procedure.

We start calculating numerically the FLTEs as a function of the iteration time n for one initial condition, $(x_0, p_0)=(-0.31, 0.0)$ chosen inside the regular island around $P_4^-$, and for three different values of dissipation parameter: $\gamma=0$ (black curve), 10⁻⁴ (red curve) and 10⁻³ (blue curve). The results are plotted in figure 8(a), where the straight horizontal line displays the value zero, for reference. The black curve (i) is the conservative case where λ tends to zero, as expected for stable periodic orbits in conservative systems. This convergency to zero can be checked for longer times in the log–log plot in the inset of figure 8(a). As dissipation increases to 10⁻⁴ the convergence of $\lambda\to 0$ in the blue curve (ii) is faster. Around $n=6\times 10^{5}$ iterations λ crosses the zero line and becomes slightly negative. For 10⁻³ this convergence, observed by the red curve (iii), is even faster and λ becomes more negative. Other values of γ were checked and the main behaviour described in figure 8(a) remains practically unchanged.

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Now we calculate the FTLEs as a function of the dissipation parameter for the same IC used to obtain figure 8(a) and plot the result in figure 8(b). For $\gamma\rightarrow 0$ we have $\lambda\rightarrow 0$, as expected for the conservative case. When the dissipation parameter slowly increases, λ becomes more and more negative in agreement with the results from figure 8(a). For $\gamma\simeq 0.26$ a peak in the value of λ is observed inside the red box drawn in figure 8(b). At the peak, the FTLE is close to $\lambda\sim 0$ suggesting that for $\gamma\simeq 0.26$ a bifurcation occurs. This is apparently the case. We remember that $\gamma\simeq 0.26$ is approximately the dissipation value for a discontinuity (or a 'jump', or 'collapse' of attractors) in the convergence point x_f (plotted in figure 5(c)). In other words, for values of $\gamma\gtrsim 0.26$, trajectories are not attracted to $P_4^\pm$ anymore, but they start to converge, via a bifurcation procedure, to the attractor at $P_1^\pm$. Thus, there is a switch of attractors via the bifurcation phenomena.

The goal of the present section is to make a systematic analysis of the collapse of higher-order to lower-order resonances as dissipation increases. For this we choose seven exemplary periodic orbits shown in figure 9(a). To better identify such orbits, for each periodic orbits we plot one surrounding irrational torus with a given colour. Forest-green around periodic point $7\times 4$, cyan around $15\times 4$, olive around $11\times 4$, dark-red around $23\times 4$, blue around $53\times 4$, thick black around $29\times 4$ and dark-violet around $19\times 4$. The choice of these periodic orbits is appropriate to analysing two main questions: (a) to which attractor and (b) in what order these orbits will collapse as dissipation increases? Notice that all periodic orbits are surrounding one of the $7\times 4$ fixed points. However, there is an important distinction between them. While periods $29\times 4, 11\times 4,$ and $15\times 4$ are trapped to $7\times 4$ by some external torus (see dark-orange torus), orbits $19\times 4, 23\times 4$ and $53\times 4$ are, naively speaking, free to be attracted to the external chaotic regime when dissipation increases. However, this is not what occurs, as shown for the orbit $23\times 4$ in figure 9(b). This periodic orbit ($P_4^{-(23)}$) is originally generated by the IC $(x_0, p_0)=(-0.386\, 885, 0.298\, 300)$ for $\gamma=0$. As dissipation increases, the $23\times 4$ orbit survives until $\gamma\simeq 0.000\, 3347$, where a bifurcation (FTLE close to zero) phenomenum occurs and it collapses directly to the $P_4^{-(7)}$ point. Thus, it does not collapse to an intermediate higher-order resonance. In fact, from the seven chosen periodic orbits, this is the only one which collapses to $P_4^{-(7)}$, all other five orbits collapse, as dissipation increases, directly to $P_4^{-(1)}$. This answers question (a), that there is apparently no way to predict exactly to which attractor higher-order resonances will collapse. In order to answer question (b) we determine the sequence, as dissipation increases, for the collapse of the periodic points directly to $P_4^{-(1)}$. This is summarized in table 1 (please note that period $23\times 4$ is not shown since it did not collapse to $P_4^{-(1)}$). The first row shows the periods of the orbits and the second row the values of γ for which these orbits collapse to $P_4^{-(1)}$. There is no evident collapse hierarchy observed in the periods of the orbits. However, a nice relation is observed when we compute the area $A_p^{(\gamma=0)}$ (third row) of the islands from the conservative case surrounding each stable point. For example, the island surrounding the period $15\times 4$ in figure 9(a) is much larger than the island surrounding period $11\times 4$. We found that the period $15\times 4$ is more stable under dissipation than $11\times 4$ (see table 1). This suggests a relation between the values of γ where collapses occur and the mentioned area. To approximately determine these areas we performed additional simulations (not shown) and estimated them directly from the phase space, in arbitrary units. We only considered the area around one orbital point from each periodic orbit. For tiny areas such as $A^{(\gamma=0)}_{29\times 4}$ and $A^{(\gamma=0)}_{53\times 4}$, we could not obtain reliable precision to distinguish them. Table 1 shows a direct relation between the increasing values of γ and the increasing values of $A_p^{(\gamma=0)}$. Therefore, the collapse of a periodic point under dissipation is proportional to the size of the island from the conservative limit surrounding these points. In other words, the collapse to $P_4^{-(1)}$ is inversely proportional to $A_p^{(\gamma=0)}$. The same results should be valid for $P_4^{+(1)}$. In addition, it is also clear that the same hierarchy occurs for the collapse of periods $7\times4$ and 4 to $P_1^{-}$. More generally, we may conjecture that $H_p(\gamma)\propto 1/A_p^{(\gamma=0)}$, where $H_p(\gamma)$ estimates the local collapse priority of a period-p stable orbit as dissipation increases and $A_p^{(\gamma=0)}$ is the area of the island from the conservative limit surrounding the corresponding periodic point.

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Table 1. Values of γ for which the stable attractors with period-p collapse to $P_4^{-(1)}$. $A_{p}^{(\gamma=0)}$ is the area, in arbitrary units, of the islands from the conservative case around the periodic point with period-p.

Period-p	$\gamma\, (\,p\to 4)$	$A_{p}^{(\gamma=0)}({\rm a.u.})$
$29\times 4$	$7.256\times 10^{-6}$	0.07
$53\times 4$	$1.422\times 10^{-5}$	0.07
$19\times 4$	$4.930\times 10^{-5}$	0.64
$11\times 4$	$5.597\times 10^{-5}$	3.99
$15\times 4$	$6.538\times 10^{-4}$	13.3