Matching of separatrix map and resonant dynamics, with application to global chaos onset between separatrices

We have developed a general method for the description of separatrix chaos, based on the analysis of the separatrix map dynamics. Matching it with the resonant Hamiltonian analysis, we show that, for a given amplitude of perturbation, the maximum width of the chaotic layer in energy may be much larger than it was assumed before. We use the above method to explain the drastic facilitation of global chaos onset in time-periodically perturbed Hamiltonian systems possessing two or more separatrices, previously discovered [S. M. Soskin, O. M. Yevtushenko, and R. Mannella, Phys. Rev. Lett. 90, 174101 (2003)]. The theory well agrees with simulations. We also discuss generalizations and applications. The method may be generalized for single-separatrix cases. The facilitation of global chaos onset may be relevant to a variety of systems, e.g., optical lattices, magnetic and semiconductor superlattices, meandering flows in the ocean, and spinning pendulums. Apart from dynamical transport, it may facilitate noise-induced transitions and the stochastic web formation.

Figure 1

Typical evolution of thin chaotic layers in the plane of slow variables of a zero-dispersion system (the perturbation magnitude grows from the top to the bottom).

Figure 2

The potential $U\left(q\right)$, the separatrices in the phase space, and the eigenfrequency $\omega \left(E\right)$ for the unperturbed system (1) with $\Phi =0.2$, in (a), (b), and (c), respectively.

Figure 3

The evolution of the stroboscopic (at $t=n2\pi ∕{\omega }_f$ with $n=0,1,2,\dots$) Poincaré section of the system (2)-(1) with $\Phi =0.2$ as $h$ grows while ${\omega }_f=0.3$. The number of points in each trajectory is 2000. In (a) and (b), three characteristic trajectories are shown: the inner trajectory starts from the state $\left\{I^{\left(0\right)}\right\}\equiv \{p=0,q=\pi ∕2\}$ and is chaotic but bounded in space; the outer trajectory starts from $\left\{O^{\left(0\right)}\right\}\equiv \{p=0,q=-\pi ∕2\}$ and is chaotic and unbounded in coordinate; the third trajectory is an example of a regular trajectory separating the two chaotic ones. In (c), the chaotic trajectories mix.

Figure 4

The diagram indicating (shaded) the perturbation parameters range for which global chaos exists. The integration time for each point of the grid is $12000\pi$.

Figure 5

(Color online) The evolution of the stroboscopic Poincaré section of the system (2)-(1) with $\Phi =0.2$ as the amplitude of the perturbation $h$ grows while the frequency is fixed at ${\omega }_f=0.401$. The number of points in each trajectory is 2000. The chaotic trajectories starting from the states $\left\{I^{\left(0\right)}\right\}$ and $\left\{O^{\left(0\right)}\right\}$ are drawn in green and blue, respectively. The stable stationary points of Eq. (14) for $n=1$ (i.e., for the first-order nonlinear resonances) are indicated by the red and cyan crosses. The chaotic layers associated with the resonances are indicated in red and cyan, respectively, unless they merge with those associated with the green or blue chaotic trajectories. Examples of regular trajectories embracing the state $\left\{I^{\left(0\right)}\right\}$ while separating various chaotic trajectories are shown in brown.

Figure 6

(Color online) The evolution of the separatrices of the first-order resonances within the resonance approximation [described by Eq. (4) with $n=1$] in the plane of action $I$ and slow angle $\tilde{\psi }$, for the same parameters as in Fig. 5. Horizontal levels mark the values of $I$ corresponding to the barriers.

Figure 7

(Color online) A schematic example illustrating the graphic solutions of Eq. (15) for $n=1$, as intersections of the curve $\omega \left(I\right)$ (thick solid red line) with the curves ${\omega }_f\pm hd{q}_n\left(I\right)∕dI$ (thin solid green lines). The solutions corresponding to the lower and upper relevant saddles [defined by Eq. (16)] are marked by dots and by the labels $sl$ and $su$, respectively (we do not mark other solutions because they are irrelevant).

Figure 8

(Color online) (a) Chaotic layers (shaded in green and blue, for the upper and lower layers, respectively) in the plane of action $I$ and slow angle $\tilde{\psi }$, as described by our theory. Parameters are the same as in Figs. 5b, 6b. The lower and upper boundaries of the figure box coincide with $I\left(E_b^{\left(1\right)}\right)$ and $I\left(E_b^{\left(2\right)}\right)$, respectively. The resonance separatrices are drawn by the cyan and red solid lines (for the lower and upper resonances, respectively). Dashed green and blue lines mark the curves $I=I_{\mathrm{GSS}}^{\left(l\right)}\left(\tilde{\psi }\right)\equiv I[E=E_b^{\left(1\right)}+{\delta }_l\mid \sin (\tilde{\psi })\mid ]$ and $I=I_{\mathrm{GSS}}^{\left(u\right)}\left(\tilde{\psi }\right)\equiv I[E=E_b^{\left(2\right)}-{\delta }_u\mid \sin (\tilde{\psi })\mid ]$, respectively, where ${\delta }_l$ and ${\delta }_u$ are the values of the separatrix split related to the lower and upper barrier, respectively. The upper boundary of the lower layer is formed by the trajectory of the resonant Hamiltonian system (4) tangent to the curve $I=I_{\mathrm{GSS}}^{\left(l\right)}\left(\tilde{\psi }\right)$. The lower boundary of the upper layer is formed by the lower part of the upper (red) resonance separatrix. The periodic closed loops (solid blue lines) are the trajectories of the system (4) tangent to the curve $I_{\mathrm{GSS}}^{\left(u\right)}\left(\tilde{\psi }\right)$: they form the boundaries of the major stability islands inside the upper chaotic layer. (b) Comparison of the chaotic layers obtained from computer simulations (dots) with the theoretically calculated boundaries (solid lines) shown in the box (a).

Figure 9

(Color online) Illustrations of the mechanisms of the formation of the first spike wings and of the corresponding theoretical lines in Fig. 10a. (a), (b), and (c) illustrate the lines by Eqs. (20, 41, 64), respectively: the corresponding perturbation parameters are $({\omega }_f=0.39,h=0.0077)$, $({\omega }_f=0.41,h=0.00598)$, and $({\omega }_f=0.43,h=0.01009)$, respectively. Resonance separatrices are drawn by red and cyan. The dashed lines show the functions $I_{\mathrm{GSS}}^{\left(l\right)}\left(\tilde{\psi }\right)$ and $I_{\mathrm{GSS}}^{\left(u\right)}\left(\tilde{\psi }\right)$. The black line in (c) is the trajectory of the resonant Hamiltonian system (4), which is tangent to both dashed lines.

Figure 10

(Color online) The first (a) and second (b) spike in $h_{gc}\left({\omega }_f\right)$: comparison between the results of the numerical simulations (the lower boundary of the shaded area) and the theoretical estimates. The estimates are indicated by the corresponding equation numbers and are drawn by different types of lines, in particular, the dashed lines represent the explicit asymptote for the solid line of the same color.

Figure 11

(Color online) Amplitude of the third Fourier harmonic as a function of action (solid red line). The dashed black line shows the zero level. Its intersection with the solid red line is marked by the circle. The green line indicates the value of action where $q_3=0$. The inset illustrates the line (66) in Fig. 10b: the GSS curve touches the horizontal line $I=I_{q_3=0}$.

Figure 12

(Color online) The universal shape of the spike minimum (67) (solid lines). The dashed line indicates the universal slope of the spike in between the minimum and the fold, which have merged in the universal (asymptotic) function (67).

Figure 13

The diagram analogous to that in Fig. 4 but for the system (71) (with ${\Phi }_0=0.2$).

Figure 14

Schematic example of the time dependence of the velocity of the perturbed system (thick solid line) in the case when the energy of motion varies in close vicinity of the top of the lower potential barrier. The dashed line marks the zero level of the velocity. Pulses of the velocity are schematically singled out by the parallelograms (drawn by a thin solid line). The two sequences of time instants $(\dots ,t_{i-1},t_i,t_{i+1},\dots )$ and $(\dots ,t_{i-1}^{\prime },t_i^{\prime },t_{i+1}^{\prime },\dots )$ correspond to the beginnings and centers of the pulses, respectively.

Figure 15

The theoretical estimates for the normalized separatrix split (for $\Phi =0.2$) as a function of the perturbation frequency, for the lower and upper layers in (a) and (b), respectively. The solid lines are calculated by Eqs. (A11, A43) [for (a) and (b), respectively] while dashed lines represent asymptotic expressions (A9, A44), respectively.

Figure 16

(Color online) A schematic figure illustrating the formation of the boundary of the lower chaotic layer for $h<h_{cr}^{\left(l\right)}\left({\omega }_f\right)$ in the ranges of ${\omega }_f$ relevant to (a) odd, and (b) even spikes. The dashed magenta line shows the GSS curve in the energy-angle plane: $E\left(\tilde{\psi }\right)=E_{\mathrm{GSS}}^{\left(l\right)}\left(\tilde{\psi }\right)\equiv E_b^{\left(1\right)}+{\delta }_l\mid \sin \left(\tilde{\psi }\right)\mid$. Green lines show examples of those trajectories (14), which have points in common with the GSS curve. One of such trajectories (14) (shown by the thick green line) relates to the formation of the upper boundary of the lower chaotic layer: in (a), the boundary is the trajectory tangent to the GSS curve; in (b), the boundary is the upper part of the separatrix generated by the saddle $s$. Yellow dots indicate the relevant common points of the GSS curve and the thick green line. They have angles $\pm {\tilde{\psi }}_t^{\left(l\right)}$ and energy $E_t^{\left(l\right)}$ in the case (a), and angles $\pm {\tilde{\psi }}_i^{\left(l\right)}$ and energy $E_i^{\left(l\right)}$ in the case (b). The minimum and maximum deviation of energy on the boundary from the barrier energy are denoted as ${\delta }_{\min }^{\left(l\right)}$ and ${\delta }_{\max }^{\left(l\right)}$, respectively. The maximum deviation on the GSS curve is equal to ${\delta }_l$.