# American Institute of Mathematical Sciences

September  2022, 21(9): 3213-3245. doi: 10.3934/cpaa.2022097

## Rank-one strange attractors versus heteroclinic tangles

 1 Center of Mathematics, University of Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal 2 Lisbon School of Economics & Management, University of Lisbon, Rua do Quelhas, 6, 1200-781 Lisbon, Portugal

Received  December 2021 Published  September 2022 Early access  June 2022

Fund Project: AR was partially supported by CMUP (UID/MAT/00144/2020), which is funded by FCT with national (MCTES) and European structural funds through the programs FEDER, under the partnership agreement PT2020. AR also acknowledges financial support from Program INVESTIGADOR FCT (IF/00107/2015)

We present two different mechanisms for the emergence of strange attractors (observable chaos) in a two-parameter periodically-perturbed family of differential equations on the plane. The two parameters are independent and act on different ways in the invariant manifolds of consecutive saddles in the cycle. When both parameters are zero, the flow exhibits an attracting heteroclinic cycle associated to two equilibria. The first parameter makes the two-dimensional invariant manifolds of consecutive saddles in the cycle to pull apart; the second forces transverse intersection. The unfolding of each parameter is associated to the emergence of different dynamical scenarios (rank-one attractors and heteroclinic tangles). These relative positions may be determined using the Melnikov method.

Extending the previous theory on the field, we prove the existence of many complicated dynamical objects in the two-parameter family, ranging from "lar-ge" strange attractors supporting SRB (Sinai-Ruelle-Bowen) measures to superstable sinks and non-uniformly hyperbolic attractors. We draw a plausible bifurcation diagram associated to the problem under consideration and we show that the occurrence of heteroclinic tangles is a prevalent phenomenon.

Citation: Alexandre A. Rodrigues. Rank-one strange attractors versus heteroclinic tangles. Communications on Pure and Applied Analysis, 2022, 21 (9) : 3213-3245. doi: 10.3934/cpaa.2022097
##### References:
 [1] V. S. Afraimovich, S-B Hsu and H. E. Lin, Chaotic behavior of three competing species of May–Leonard model under small periodic perturbations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 435-447.  doi: 10.1142/S021812740100216X. [2] M. A. D. Aguiar, S. B. S. D. Castro and I. S. Labouriau, Dynamics near a heteroclinic network, Nonlinearity, 18 (2005), 391-414.  doi: 10.1088/0951-7715/18/1/019. [3] A. L. Bertozzi, Heteroclinic orbits and chaotic dynamics in planar fluid flows, SIAM J. Math. Anal., 19 (1988), 1271-1294.  doi: 10.1137/0519093. [4] H. Broer, C. Simó and J. C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms, Nonlinearity, 11 (1998), 667-770.  doi: 10.1088/0951-7715/11/3/015. [5] F. Chen, A. Oksasoglu and Q. Wang, Heteroclinic tangles in time-periodic equations, J. Diff. Eqs., 254 (2013), 1137-1171.  doi: 10.1016/j.jde.2012.10.010. [6] S-N. Chow and J. Hale, Methods of Bifurcation Theory, Grundlehren der mathematischen Wissenschaften, Springer-Verlag New York-Berlin, 1982. doi: 10.1007/978-1-4613-8159-4. [7] N. K. Gavrilov and L. P. Shilnikov, On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve–Part II, Math. USSR Sbornik, 254 (1973), 139-156.  doi: 10.1070/SM1972v017n04ABEH001597. [8] J. Guckenheimer and P. Holmes,, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Science, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-1140-2. [9] M. Herman, Mesure de Lebesgue et Nombre de Rotation, Lecture Notes in Math., 597 (1977), 271–293. doi: 10.1007/BFb0085359. [10] I. S. Labouriau and A. A. P. Rodrigues, On Takens' Last Problem: tangencies and time averages near heteroclinic networks, Nonlinearity, 30 (2017), 1876-1910.  doi: 10.1088/1361-6544/aa64e9. [11] V. K. Melnikov, On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc., 12 (1963), 1-57. [12] A. Mohapatra and W. Ott, Homoclinic Loops, Heteroclinic Cycles, and Rank One Dynamics, SIAM J. Appl. Dyn. Syst., 14 (2015), 107-131.  doi: 10.1137/140995659. [13] L. Mora and M. Viana, Abundance of strange attractors, Acta Math., 171 (1993), 1-71.  doi: 10.1007/BF02392766. [14] S. E. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., 50 (1979), 101-151.  doi: 10.1007/BF02684771. [15] W. Ott and Q. Wang, Periodic attractors versus nonuniform expansion in singular limits of families of rank one maps, Discrete Contin. Dyn. Syst., 26 (2010), 1035-1054.  doi: 10.3934/dcds.2010.26.1035. [16] H. Poincaré, Les méthodes nouvelles de la mécanique céleste - Tome III, New methods of celestial mechanics. Vol. III, 26 Gauthier-Villars: Paris, (1899). [17] A. A.P. Rodrigues, "Large" strange attractors in the unfolding of a heteroclinic attractor, Discrete Contin. Dyn. Syst., 42 (2022), 2355-2379.  doi: 10.3934/dcds.2021193. [18] A. A. P. Rodrigues, I. S. Labouriau and M. A. D. Aguiar, Chaotic double cycling, Dyn. Sys. Int. J., 26 (2011), 199-233.  doi: 10.1080/14689367.2011.557179. [19] L. P. Shilnikov, On the question of the structure of an extended neighborhood of a structurally stable state of equilibrium of saddle-focus type, Math. USSR Sb., 81 (1970), 92-103.  doi: 10.1070/SM1970v010n01ABEH001588. [20] L. P. Shilnikov, A. Shilnikov, D. Turaev and L. Chua, Methods of Qualitative Theory In Nonlinear Dynamics–Part II, World Sci. Singapore, New Jersey, London, Hong Kong, 2001. doi: 10.1142/4221. [21] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 474-517. [22] H. Takahasi and Q. Wang., Nonuniformly expanding 1d maps with logarithmic singularities, Nonlinearity, 25 (2012), 533-550.  doi: 10.1088/0951-7715/25/2/533. [23] F. Takens, Heteroclinic attractors: time averages and moduli of topological conjugacy, Bull. Braz. Math. Soc., 25 (1994), 107-120.  doi: 10.1137/0152085. [24] Q. Wang and A. Oksasoglu, Dynamics of homoclinic tangles in periodically perturbed second-order equations, J. Diff. Eqs., 250 (2011), 710-751.  doi: 10.1016/j.jde.2010.04.005. [25] Q. Wang and W. Ott, Dissipative homoclinic loops of two-dimensional maps and strange attractors with one direction of instability, Comm. Pure Appl. Math., 64 (2011), 1439-1496.  doi: 10.1002/CPA.20379. [26] Q. Wang and L.-S. Young, From invariant curves to strange attractors, Commun. Math. Phys., 225 (2002), 225-275.  doi: 10.1007/s002200100582. [27] Q. Wang and L.-S. Young, Strange attractors in periodically-kicked limit cycles and Hopf bifurcations, Commun. Math. Phys., 240 (2003), 509-529.  doi: 10.1007/s00220-003-0902-9. [28] Q. Wang and L.-S. Young, Nonuniformly expanding 1D maps, Commun. Math. Phys., 264 (2006), 255-282.  doi: 10.1007/s00220-005-1485-4.

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##### References:
 [1] V. S. Afraimovich, S-B Hsu and H. E. Lin, Chaotic behavior of three competing species of May–Leonard model under small periodic perturbations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 435-447.  doi: 10.1142/S021812740100216X. [2] M. A. D. Aguiar, S. B. S. D. Castro and I. S. Labouriau, Dynamics near a heteroclinic network, Nonlinearity, 18 (2005), 391-414.  doi: 10.1088/0951-7715/18/1/019. [3] A. L. Bertozzi, Heteroclinic orbits and chaotic dynamics in planar fluid flows, SIAM J. Math. Anal., 19 (1988), 1271-1294.  doi: 10.1137/0519093. [4] H. Broer, C. Simó and J. C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms, Nonlinearity, 11 (1998), 667-770.  doi: 10.1088/0951-7715/11/3/015. [5] F. Chen, A. Oksasoglu and Q. Wang, Heteroclinic tangles in time-periodic equations, J. Diff. Eqs., 254 (2013), 1137-1171.  doi: 10.1016/j.jde.2012.10.010. [6] S-N. Chow and J. Hale, Methods of Bifurcation Theory, Grundlehren der mathematischen Wissenschaften, Springer-Verlag New York-Berlin, 1982. doi: 10.1007/978-1-4613-8159-4. [7] N. K. Gavrilov and L. P. Shilnikov, On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve–Part II, Math. USSR Sbornik, 254 (1973), 139-156.  doi: 10.1070/SM1972v017n04ABEH001597. [8] J. Guckenheimer and P. Holmes,, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Science, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-1140-2. [9] M. Herman, Mesure de Lebesgue et Nombre de Rotation, Lecture Notes in Math., 597 (1977), 271–293. doi: 10.1007/BFb0085359. [10] I. S. Labouriau and A. A. P. Rodrigues, On Takens' Last Problem: tangencies and time averages near heteroclinic networks, Nonlinearity, 30 (2017), 1876-1910.  doi: 10.1088/1361-6544/aa64e9. [11] V. K. Melnikov, On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc., 12 (1963), 1-57. [12] A. Mohapatra and W. Ott, Homoclinic Loops, Heteroclinic Cycles, and Rank One Dynamics, SIAM J. Appl. Dyn. Syst., 14 (2015), 107-131.  doi: 10.1137/140995659. [13] L. Mora and M. Viana, Abundance of strange attractors, Acta Math., 171 (1993), 1-71.  doi: 10.1007/BF02392766. [14] S. E. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., 50 (1979), 101-151.  doi: 10.1007/BF02684771. [15] W. Ott and Q. Wang, Periodic attractors versus nonuniform expansion in singular limits of families of rank one maps, Discrete Contin. Dyn. Syst., 26 (2010), 1035-1054.  doi: 10.3934/dcds.2010.26.1035. [16] H. Poincaré, Les méthodes nouvelles de la mécanique céleste - Tome III, New methods of celestial mechanics. Vol. III, 26 Gauthier-Villars: Paris, (1899). [17] A. A.P. Rodrigues, "Large" strange attractors in the unfolding of a heteroclinic attractor, Discrete Contin. Dyn. Syst., 42 (2022), 2355-2379.  doi: 10.3934/dcds.2021193. [18] A. A. P. Rodrigues, I. S. Labouriau and M. A. D. Aguiar, Chaotic double cycling, Dyn. Sys. Int. J., 26 (2011), 199-233.  doi: 10.1080/14689367.2011.557179. [19] L. P. Shilnikov, On the question of the structure of an extended neighborhood of a structurally stable state of equilibrium of saddle-focus type, Math. USSR Sb., 81 (1970), 92-103.  doi: 10.1070/SM1970v010n01ABEH001588. [20] L. P. Shilnikov, A. Shilnikov, D. Turaev and L. Chua, Methods of Qualitative Theory In Nonlinear Dynamics–Part II, World Sci. Singapore, New Jersey, London, Hong Kong, 2001. doi: 10.1142/4221. [21] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 474-517. [22] H. Takahasi and Q. Wang., Nonuniformly expanding 1d maps with logarithmic singularities, Nonlinearity, 25 (2012), 533-550.  doi: 10.1088/0951-7715/25/2/533. [23] F. Takens, Heteroclinic attractors: time averages and moduli of topological conjugacy, Bull. Braz. Math. Soc., 25 (1994), 107-120.  doi: 10.1137/0152085. [24] Q. Wang and A. Oksasoglu, Dynamics of homoclinic tangles in periodically perturbed second-order equations, J. Diff. Eqs., 250 (2011), 710-751.  doi: 10.1016/j.jde.2010.04.005. [25] Q. Wang and W. Ott, Dissipative homoclinic loops of two-dimensional maps and strange attractors with one direction of instability, Comm. Pure Appl. Math., 64 (2011), 1439-1496.  doi: 10.1002/CPA.20379. [26] Q. Wang and L.-S. Young, From invariant curves to strange attractors, Commun. Math. Phys., 225 (2002), 225-275.  doi: 10.1007/s002200100582. [27] Q. Wang and L.-S. Young, Strange attractors in periodically-kicked limit cycles and Hopf bifurcations, Commun. Math. Phys., 240 (2003), 509-529.  doi: 10.1007/s00220-003-0902-9. [28] Q. Wang and L.-S. Young, Nonuniformly expanding 1D maps, Commun. Math. Phys., 264 (2006), 255-282.  doi: 10.1007/s00220-005-1485-4.
The dynamics of $(2.1)$ defined $\mathcal{V}\subset {\mathbb{R}}^2$ is governed by the existence of an heteroclinic cycle associated to $O_1$ and $O_2$. $\ell_1, \ell_2$: heteroclinic connections; $\mathcal{V}^\star$: inner basin of attraction of the cycle (absorbing domain); $\mathcal{A}:$ region limited by the cycle
(a) Scheme of the effects of the parameters $\mu_1, \mu_2$ on the equations (2.3). (b) Sketch of the local and transition maps. $I, II, III$ and $IV$ represent cross sections ${\text{Out}}( \textbf{C}_1)$, ${\text{In}}( \textbf{C}_2)$, ${\text{Out}}( \textbf{C}_2)$ and ${\text{In}}( \textbf{C}_1)$, respectively
Plausible bifurcation diagram associated to an element $f_{(\mu_1, \mu_2)}$ of the family $\mathfrak{X}_{\Gamma}^4(\mathbb{V})$. I – the flow has an invariant two-dimensional torus if $\omega\approx 0$ and a rank-one strange attractor if $\omega \gg 1$. II – transition region; III – heteroclinic tangle; Hom – curve that corresponds to the emergence of a homoclinic tangency associated to $\textbf{C}_1$
Example of a Misiurewicz-type map $h_a: {\mathbb{S}}^1 \rightarrow {\mathbb{R}}$. For $\delta>0$, the set $C_{\delta}$ is a $\delta$-neighbourhood of the set of critical points $C$
Transition maps from ${\text{Out}}( \textbf{C}_1)$ to ${\text{In}} ( \textbf{C}_2)$ (Case I) and from ${\text{Out}}( \textbf{C}_2)$ to ${\text{In}} ( \textbf{C}_1)$ (Case II). In Case I: $W^u( \textbf{C}_1) \cap W^s( \textbf{C}_2) = \emptyset$. In Case II: $W^u( \textbf{C}_2) \pitchfork W^s( \textbf{C}_1)$. Double bars mean that the sides are identified
Graph of the map $h_a$ for $a = \mu_n$ and $\omega \gg 1$ with $q = 2$ (number of critical points). Indicated is a superstable periodic orbit of period 2
The set $\eta_u = W^u( \textbf{C}_2)\cap {\text{In}}^+ ( \textbf{C}_1)$ is mapped by ${Loc}_2\circ \Psi_{1\rightarrow 2}\circ Loc_1$ into a spiral accumulating on the circle defined by $\overline{y}_1^{(2)} = 0$ (ie the parametrisation of ${\text{Out}}( \textbf{C}_2)\cap W^u( \textbf{C}_2)$). There is a sequence $(\mu_i)_i$ for which the flow of $\mathcal{G}_{\mu_i}$ exhibits a quadratic heteroclinic tangency
Four types of configuration for (2.2) when $\mu_1>0$ and $\mu_2 = 0$. (a), (b): one contractible periodic solution; (c): two contractible periodic solutions; (d) one non-contractible periodic solution. Double bars mean that the sides are identified
Four generic unfoldings for the dynamics of (2.3)
 Configuration $W^u( \textbf{C}_1) \pitchfork W^s( \textbf{C}_2)$ $W^u( \textbf{C}_1) \cap W^s( \textbf{C}_2)=\emptyset$ $W^u( \textbf{C}_2) \pitchfork W^s( \textbf{C}_1)$ Case 1 Case 2 $W^u( \textbf{C}_2) \cap W^s( \textbf{C}_1)=\emptyset$ Case 3 Case 4
 Configuration $W^u( \textbf{C}_1) \pitchfork W^s( \textbf{C}_2)$ $W^u( \textbf{C}_1) \cap W^s( \textbf{C}_2)=\emptyset$ $W^u( \textbf{C}_2) \pitchfork W^s( \textbf{C}_1)$ Case 1 Case 2 $W^u( \textbf{C}_2) \cap W^s( \textbf{C}_1)=\emptyset$ Case 3 Case 4
Coordinates and notation of the cross sections after the change of coordinates (6.1) and (6.2)
 Set Notation Set Notation ${\text{In}} ( \textbf{C}_1)$ $\left(y_1^{(1)}, y_2^{(1)}, \theta^{(1)}\right)$ ${\text{In}} ( \textbf{C}_2)$ $\left(y_1^{(2)}, y_2^{(2)}, \theta^{(2)}\right)$ ${\text{Out}} ( \textbf{C}_1)$ $\left(\overline{y}_1^{(1)}, \overline{y}_2^{(1)}, \overline{\theta}^{(1)}\right)$ ${\text{Out}} ( \textbf{C}_2)$ $\left(\overline{y}_1^{(2)}, \overline{y}_2^{(2)}, \overline{\theta}^{(2)}\right)$
 Set Notation Set Notation ${\text{In}} ( \textbf{C}_1)$ $\left(y_1^{(1)}, y_2^{(1)}, \theta^{(1)}\right)$ ${\text{In}} ( \textbf{C}_2)$ $\left(y_1^{(2)}, y_2^{(2)}, \theta^{(2)}\right)$ ${\text{Out}} ( \textbf{C}_1)$ $\left(\overline{y}_1^{(1)}, \overline{y}_2^{(1)}, \overline{\theta}^{(1)}\right)$ ${\text{Out}} ( \textbf{C}_2)$ $\left(\overline{y}_1^{(2)}, \overline{y}_2^{(2)}, \overline{\theta}^{(2)}\right)$
Notation for the next sections where $\mu_1, \mu_2\in\, \, ]0, \varepsilon]$ with $\varepsilon$ is small
 $\mathcal{F}_{(\mu_1, 0)} \mapsto \mathcal{F}_{\mu}$ Sections 7 and 8 $\mathcal{G}_{(0, \mu_2)}\mapsto \mathcal{G}_{\mu}$ Sections 9 and 10
 $\mathcal{F}_{(\mu_1, 0)} \mapsto \mathcal{F}_{\mu}$ Sections 7 and 8 $\mathcal{G}_{(0, \mu_2)}\mapsto \mathcal{G}_{\mu}$ Sections 9 and 10
Overview of the results in the literature and the contribution of the present article (in blue) for the dynamics of (2.3)
 Configuration $W^u( \textbf{C}_1) \pitchfork W^s( \textbf{C}_2)$ $W^u( \textbf{C}_1) \cap W^s( \textbf{C}_2)=\emptyset$ $W^u( \textbf{C}_2) \pitchfork W^s( \textbf{C}_1)$ Case 1Heteroclinic tangles(Horseshoes, tangencies, sinksNewhouse phenomena, pulses)[5,10,18] Case 2 (Novelty of this article)($\mu_1>0$ and $\mu_2=0$)Region with torus if $\omega \approx 0$Region with rank-one attractors if $\omega \gg 1$Superstable sinks if $\omega \gg 1$($\mu_1=0$ and $\mu_2>0$)Non-uniformly hyperbolic strange attractorsif $\omega \gg 1$($\mu_1, \mu_2>0$)Heteroclinic tangles prevail $W^u( \textbf{C}_2) \cap W^s( \textbf{C}_1)=\emptyset$ Case 3Similar to Case 2 Case 4Region with torus if $\omega \approx 0$Region with rank-one attractors if $\omega \gg 1$[12]
 Configuration $W^u( \textbf{C}_1) \pitchfork W^s( \textbf{C}_2)$ $W^u( \textbf{C}_1) \cap W^s( \textbf{C}_2)=\emptyset$ $W^u( \textbf{C}_2) \pitchfork W^s( \textbf{C}_1)$ Case 1Heteroclinic tangles(Horseshoes, tangencies, sinksNewhouse phenomena, pulses)[5,10,18] Case 2 (Novelty of this article)($\mu_1>0$ and $\mu_2=0$)Region with torus if $\omega \approx 0$Region with rank-one attractors if $\omega \gg 1$Superstable sinks if $\omega \gg 1$($\mu_1=0$ and $\mu_2>0$)Non-uniformly hyperbolic strange attractorsif $\omega \gg 1$($\mu_1, \mu_2>0$)Heteroclinic tangles prevail $W^u( \textbf{C}_2) \cap W^s( \textbf{C}_1)=\emptyset$ Case 3Similar to Case 2 Case 4Region with torus if $\omega \approx 0$Region with rank-one attractors if $\omega \gg 1$[12]
Notation
 Notation Definition/meaning Section $\mathcal{V}$ Open region of ${\mathbb{R}}^2$ where equation (2.1) is well defined § 2.1 $O_1, O_2$ Saddle-equilibria of the equation (2.1) § 2.1 ${\ell}_1, {\ell}_2$ Connections from $O_1$ to $O_2$ and from $O_2$ to $O_1$ § 2.1 $\mathcal{A}$ Region limited by the heteroclinic cycle ${\ell}_1\cup {\ell}_2$ § 2.1 $\mathcal{V}^\star$ Inner basin of attraction of the heteroclinic cycle ${\ell}_1\cup {\ell}_2$(absorbing domain) § 2.1 $\mathbb{V}$ $\mathcal{V}\times {\mathbb{S}}^1$ – open region where equation (2.3) is defined § 2.2 $\textbf{C}_1, \textbf{C}_2$ Saddle periodic solutions of the equation (2.3) § 2.2 $\Gamma$ Heteroclinic cycle associated to $\textbf{C}_1, \textbf{C}_2$ § 2.2 $\mathbb{A}$ $\mathcal{A}\times {\mathbb{S}}^1$ § 2.2 $\mathbb{V}^\star$ $\mathcal{V}^\star\times {\mathbb{S}}^1$ § 2.2 $\mathcal{L}_1, \mathcal{L}_2$ Connections from $\textbf{C}_1$ to $\textbf{C}_2$ and from $\textbf{C}_2$ to $\textbf{C}_1$ § 2.2 $V_1, V_2$ Hollow cylinders around $\textbf{C}_1$ and $\textbf{C}_2$ § 2.2 $A\equiv_\mathbb{V^\star} B$ The manifolds $A$ and $B$ coincide within $\mathbb{V}^\star$ § 2.3 $\mathcal{F}_{(\mu_1, \mu_2)}\equiv \mathcal{F}_{\mu}$ First return map to ${\text{Out}}( \textbf{C}_1)$ § 6.5 andTable 3 $\mathcal{G}_{(\mu_1, \mu_2)}\equiv \mathcal{G}_{\mu}$ First return map to ${\text{Out}}( \textbf{C}_2)$ § 6.5 andTable 3
 Notation Definition/meaning Section $\mathcal{V}$ Open region of ${\mathbb{R}}^2$ where equation (2.1) is well defined § 2.1 $O_1, O_2$ Saddle-equilibria of the equation (2.1) § 2.1 ${\ell}_1, {\ell}_2$ Connections from $O_1$ to $O_2$ and from $O_2$ to $O_1$ § 2.1 $\mathcal{A}$ Region limited by the heteroclinic cycle ${\ell}_1\cup {\ell}_2$ § 2.1 $\mathcal{V}^\star$ Inner basin of attraction of the heteroclinic cycle ${\ell}_1\cup {\ell}_2$(absorbing domain) § 2.1 $\mathbb{V}$ $\mathcal{V}\times {\mathbb{S}}^1$ – open region where equation (2.3) is defined § 2.2 $\textbf{C}_1, \textbf{C}_2$ Saddle periodic solutions of the equation (2.3) § 2.2 $\Gamma$ Heteroclinic cycle associated to $\textbf{C}_1, \textbf{C}_2$ § 2.2 $\mathbb{A}$ $\mathcal{A}\times {\mathbb{S}}^1$ § 2.2 $\mathbb{V}^\star$ $\mathcal{V}^\star\times {\mathbb{S}}^1$ § 2.2 $\mathcal{L}_1, \mathcal{L}_2$ Connections from $\textbf{C}_1$ to $\textbf{C}_2$ and from $\textbf{C}_2$ to $\textbf{C}_1$ § 2.2 $V_1, V_2$ Hollow cylinders around $\textbf{C}_1$ and $\textbf{C}_2$ § 2.2 $A\equiv_\mathbb{V^\star} B$ The manifolds $A$ and $B$ coincide within $\mathbb{V}^\star$ § 2.3 $\mathcal{F}_{(\mu_1, \mu_2)}\equiv \mathcal{F}_{\mu}$ First return map to ${\text{Out}}( \textbf{C}_1)$ § 6.5 andTable 3 $\mathcal{G}_{(\mu_1, \mu_2)}\equiv \mathcal{G}_{\mu}$ First return map to ${\text{Out}}( \textbf{C}_2)$ § 6.5 andTable 3
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