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Predicting the separation of time scales in a heteroclinic network

Maximilian Voit
Maximilian Voit
 and 
Hildegard Meyer-Ortmanns
Hildegard Meyer-Ortmanns
   | Jun 28, 2019
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Article Category: Papers dedicated to the memory of Valetnin Afraymovich (1945-2018)
Published Online: Jun 28, 2019
Page range: 279 - 288
Received: Nov 26, 2019
Accepted: May 03, 2019
DOI: https://doi.org/10.2478/AMNS.2019.1.00024
Keywords
© 2019 Maximilian Voit, Hildegard Meyer-Ortmanns, published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Sketch of the topology of the hierarchical heteroclinic network. Black and grey dashed lines mark heteroclinic orbits with arrows indicating their directions. The red solid line gives an example for a trajectory in the vicinity of the network.
Sketch of the topology of the hierarchical heteroclinic network. Black and grey dashed lines mark heteroclinic orbits with arrows indicating their directions. The red solid line gives an example for a trajectory in the vicinity of the network.

Fig. 2

(a) Schematic sketch of the vicinity of saddle σ1, including the cross-sections studied in section 3 together with the local maps ϕ312 and ϕ314 and the global map ψ12. (b) Sketch of phase space near the saddle σ1 showing contracting small (r3), expanding small (r2), and radial (r1) directions. The solid line gives an example of a trajectory in the vicinity of the heteroclinic network. Dashed lines mark the heteroclinic orbits. While the deviation x is highlighted in this projection, y0 cannot be visualized without the r4 direction.
(a) Schematic sketch of the vicinity of saddle σ1, including the cross-sections studied in section 3 together with the local maps ϕ312 and ϕ314 and the global map ψ12. (b) Sketch of phase space near the saddle σ1 showing contracting small (r3), expanding small (r2), and radial (r1) directions. The solid line gives an example of a trajectory in the vicinity of the heteroclinic network. Dashed lines mark the heteroclinic orbits. While the deviation x is highlighted in this projection, y0 cannot be visualized without the r4 direction.

Fig. 3

(a) Plot of the coordinates x = res(i) and yi = rel(i) of a real trajectory when passing Hiin,cs(i)∀i∈1,2,3$H_{i}^{\operatorname{in},cs\left( i \right)}\forall i\in 1,2,3$(red⊙).$\left. \odot \right).$The dynamics switches to the subsequent SHC when the line xFE${{x}^{\frac{F}{E}}}$is passed after the fifth iteration of the return map g. Blue × mark the analytical prediction starting from point “0” assuming A = 1 = B in eq. (19). When the actual coefficients A = 1.38,B = 2.42 (read off from the numerical data when the trajectory return to the section Hiin,cs(i)$H_{i}^{\operatorname{in},cs\left( i \right)}$for the first time) are initially inserted in eq. (19), which then is iteratively applied, the analytical predictions lie precisely on top of the numerically obtained values. The small red circles below the larger ones indicate a visit of the second and third saddle within the respective SHCs. (b) Same plot as in panel (a), now including also numerical values of switches between saddles at the next SHC (green ⚀), together with the transition leading there (green ◊$\left.\lozenge \right.$).
(a) Plot of the coordinates x = res(i) and yi = rel(i) of a real trajectory when passing Hiin,cs(i)∀i∈1,2,3$H_{i}^{\operatorname{in},cs\left( i \right)}\forall i\in 1,2,3$(red⊙).$\left. \odot \right).$The dynamics switches to the subsequent SHC when the line xFE${{x}^{\frac{F}{E}}}$is passed after the fifth iteration of the return map g. Blue × mark the analytical prediction starting from point “0” assuming A = 1 = B in eq. (19). When the actual coefficients A = 1.38,B = 2.42 (read off from the numerical data when the trajectory return to the section Hiin,cs(i)$H_{i}^{\operatorname{in},cs\left( i \right)}$for the first time) are initially inserted in eq. (19), which then is iteratively applied, the analytical predictions lie precisely on top of the numerically obtained values. The small red circles below the larger ones indicate a visit of the second and third saddle within the respective SHCs. (b) Same plot as in panel (a), now including also numerical values of switches between saddles at the next SHC (green ⚀), together with the transition leading there (green ◊$\left.\lozenge \right.$).
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