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Fractal basin boundaries and escape dynamics in a multiwell potential

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Abstract

The escape dynamics in a two-dimensional multiwell potential is explored. A thorough numerical investigation is conducted in several types of two-dimensional planes and also in a three-dimensional subspace of the entire four-dimensional phase space in order to distinguish between non-escaping (ordered and chaotic) and escaping orbits. The determination of the location of the basins of escape toward the different escape channels and their correlations with the corresponding escape time of the orbits is undoubtedly an issue of paramount importance. It was found that in all examined cases regions of non-escaping motion coexist with several basins of escape. Furthermore, we monitor how the percentages of all types of orbits evolve when the total orbital energy varies. The larger escape periods have been measured for orbits with initial conditions in the fractal basin boundaries, while the lowest escape rates belong to orbits with initial conditions inside the basins of escape. The Newton–Raphson basins of attraction of the equilibrium points of the system have also been determined. We hope that our numerical analysis will be useful for a further understanding of the escape mechanism of orbits in open Hamiltonian systems with two degrees of freedom.

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Notes

  1. The SALI method was chosen over more classical dynamical indicators (e.g., the positive Lyapunov exponent, the fast Fourier transform) for distinguishing between order and chaos because it can automatically classify initial conditions of orbits using only the final numerical value of SALI at the end of the numerical integration. On the other hand, for all other classical methods we need to plot either the time evolution of the indicator (e.g., the positive Lyapunov exponent) or the shape of the spectrum (e.g., the fast Fourier transform) in order to determine the character of an orbit. Obviously, this is not possible when we have to classify large sets of initial conditions of orbits. In this case, the ideal solution is a “one-number index” such as the SALI.

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Acknowledgments

I would like to express my warmest thanks to the four anonymous referees for the careful reading of the manuscript and for all the apt suggestions and comments which allowed us to improve both the quality and the clarity of the paper.

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  1. Department of Physics, School of Science, Aristotle University of Thessaloniki, 541 24, Thessaloniki, Greece

    Euaggelos E. Zotos

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Appendix: Derivation of the multivariate Newton–Raphson iterative scheme

Appendix: Derivation of the multivariate Newton–Raphson iterative scheme

The multivariate Newton–Raphson method iterates over the recursive formula

$$\begin{aligned} \mathbf{{x}}_{n+1} = \mathbf{{x}}_{n} - J^{-1}f(\mathbf{{x}}_{n}), \end{aligned}$$
(14)

where \(J^{-1}\) is the inverse Jacobian matrix of \(f(\mathbf{{x}}_{n})\), while \(\mathbf{{x}}_{n}\) stands for the vector x at the n-th iteration. In our case the system of the equations is \(V_x = 0\) and \(V_y = 0\) and therefore the Jacobian matrix reads

$$\begin{aligned} J = \begin{bmatrix} V_{xx}&V_{xy} \\ V_{yx}&V_{yy} \end{bmatrix}. \end{aligned}$$
(15)

The inverse Jacobian is

$$\begin{aligned} J^{-1} = \frac{1}{\mathrm{{det}}(J)} \begin{bmatrix} V_{yy}&-V_{xy} \\ -V_{yx}&V_{xx} \end{bmatrix}, \end{aligned}$$
(16)

where \(\mathrm{{det}}(J) = V_{yy} V_{xx} - V_{xy}^2\).

Inserting the expression of the inverse Jacobian into the iterative formula (14), we get

$$\begin{aligned}&\begin{bmatrix} x \\ y \end{bmatrix} _{n+1}\nonumber \\&\quad = \begin{bmatrix} x \\ y \end{bmatrix} _{n} - \frac{1}{V_{yy} V_{xx} - V_{xy}^2} \begin{bmatrix} V_{yy}&-V_{xy} \\ -V_{yx}&V_{xx} \end{bmatrix} \begin{bmatrix} V_x \\ V_y \end{bmatrix} _{(x_n,y_n)} \nonumber \\&\quad = \begin{bmatrix} x \\ y \end{bmatrix} _{n} - \frac{1}{V_{yy} V_{xx} - V_{xy}^2} \begin{bmatrix} V_{yy}V_x - V_{xy}V_y \\ -V_{yx}V_x + V_{xx}V_y \end{bmatrix} _{(x_n,y_n)}.\nonumber \\ \end{aligned}$$
(17)

Decomposing formula (17) into x and y, we obtain the iterative formulae for each coordinate

$$\begin{aligned} x_{n+1}= & {} x_n - \left( \frac{V_x V_{yy} - V_y V_{xy}}{V_{yy} V_{xx} - V^2_{xy}} \right) _{(x_n,y_n)}, \nonumber \\ y_{n+1}= & {} y_n + \left( \frac{V_x V_{yx} - V_y V_{xx}}{V_{yy} V_{xx} - V^2_{xy}} \right) _{(x_n,y_n)}. \end{aligned}$$
(18)

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Zotos, E.E. Fractal basin boundaries and escape dynamics in a multiwell potential. Nonlinear Dyn 85, 1613–1633 (2016). https://doi.org/10.1007/s11071-016-2782-5

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