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November  2019, 39(11): 6541-6564. doi: 10.3934/dcds.2019284

Moduli of stability for heteroclinic cycles of periodic solutions

1. 

Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal

2. 

Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany

Received  January 2019 Revised  June 2019 Published  August 2019

We consider $ C^2 $ vector fields in the three dimensional sphere with an attracting heteroclinic cycle between two periodic hyperbolic solutions with real Floquet multipliers. The proper basin of this attracting set exhibits historic behavior and from the asymptotic properties of its orbits we obtain a complete set of invariants under topological conjugacy in a neighborhood of the cycle. As expected, this set contains the periods of the orbits involved in the cycle, a combination of their angular speeds, the rates of expansion and contraction in linearizing neighborhoods of them, besides information regarding the transition maps and the transition times between these neighborhoods. We conclude with an application of this result to a class of cycles obtained by the lifting of an example of R. Bowen.

Citation: Maria Carvalho, Alexander Lohse, Alexandre A. P. Rodrigues. Moduli of stability for heteroclinic cycles of periodic solutions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6541-6564. doi: 10.3934/dcds.2019284
References:
[1]

M. A. D. AguiarS. B. S. D. Castro and I. S. Labouriau, Simple vector fields with complex behavior, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 369-381.  doi: 10.1142/S021812740601485X.  Google Scholar

[2]

V. Arnold, V. Afraimovich, Y. Ilyashenko and L. P. Shilnikov, Bifurcation Theory and Catastrophe Theory, Encyclopaedia of Mathematical Sciences, 5. V. I. Arnold (ed.), Springer-Verlag, Berlin, 1994.  Google Scholar

[3]

J. A. Beloqui, Modulus of stability for vector fields on 3-manifolds, J. Differential Equations, 65 (1986), 374-395.  doi: 10.1016/0022-0396(86)90025-2.  Google Scholar

[4]

C. Bonatti and E. Dufraine, Éuivalence topologique de connexions de selles en dimension 3, (French) [Topological equivalence of connections between 3-dimensional saddles], Ergodic Theory Dynam. Systems, 23 (2003), 1347-1381.  doi: 10.1017/S0143385703000130.  Google Scholar

[5]

M. Carvalho and A. P. Rodrigues, Complete set of invariants for a Bykov attractor, Regul. Chaotic. Dyn., 23 (2018), 227-247.  doi: 10.1134/S1560354718030012.  Google Scholar

[6]

P. ChossatM. Golubitsky and B. L. Keyfitz, Hopf-Hopf mode interactions with O(2) symmetry, Dynam. Stability Systems, 1 (1986), 255-292.  doi: 10.1080/02681118608806019.  Google Scholar

[7]

E. Dufraine, Some topological invariants for three-dimensional flows, Chaos, 11 (2011), 443-448.  doi: 10.1063/1.1385918.  Google Scholar

[8]

M. J. Field, Equivariant dynamical systems, Trans. Amer. Math. Soc., 259 (1980), 185-205.  doi: 10.1090/S0002-9947-1980-0561832-4.  Google Scholar

[9]

M. Krupa and I. Melbourne, Asymptotic stability of heteroclinic cycles in systems with symmetry, Ergodic Theory Dynam. Systems, 15 (1995), 121-147.  doi: 10.1017/S0143385700008270.  Google Scholar

[10]

M. Krupa and I. Melbourne, Asymptotic stability of heteroclinic cycles in systems with symmetry. II, Proc. Roy. Soc. Edinburg Sect. A, 134 (2004), 1177-1197.  doi: 10.1017/S0308210500003693.  Google Scholar

[11]

I. S. Labouriau and A. A. P. Rodrigues, Dense heteroclinic tangencies near a Bykov cycle, J. Differential Equations, 259 (2015), 5875-5902.  doi: 10.1016/j.jde.2015.07.017.  Google Scholar

[12]

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.  Google Scholar

[13]

I. Melbourne, Intermittency as a codimension-three phenomenon, J. Dynam. Differential Equations, 1 (1989), 347-367.  doi: 10.1007/BF01048454.  Google Scholar

[14]

J. Palis, A differentiable invariant of topological conjugacies and moduli of stability, Astérisque, 51 (1987), 335-346.   Google Scholar

[15]

A. A. P. Rodrigues, Moduli for heteroclinic connections involving Saddle-Foci and periodic solutions, Discrete Continuous Dynam. Systems, 35 (2015), 3155-3182.  doi: 10.3934/dcds.2015.35.3155.  Google Scholar

[16]

A. A. P. RodriguesI. S. Labouriau and M. A. D. Aguiar, Chaotic double cycling, Dynamical Systems: An International Journal., 26 (2011), 199-233.  doi: 10.1080/14689367.2011.557179.  Google Scholar

[17]

D. Ruelle, Historic behaviour in smooth dynamical systems, Global Analysis of Dynamical Systems, Institute of Physics Publishing, Bristol, (2001), 63–66.  Google Scholar

[18]

A. Susín and C. Simó, On moduli of conjugation for some n-dimensional vector fields, J. Differential Equations, 79 (1989), 168-177.  doi: 10.1016/0022-0396(89)90118-6.  Google Scholar

[19]

F. Takens, Partially hyperbolic fixed points, Topology, 10 (1971), 133-147.  doi: 10.1016/0040-9383(71)90035-8.  Google Scholar

[20]

F. Takens, Heteroclinic attractors: Time averages and moduli of topological conjugacy, Bol. Soc. Brasil. Mat. (N.S.), 25 (1994), 107-120.  doi: 10.1007/BF01232938.  Google Scholar

[21]

Y. Togawa, A modulus of 3-dimensional vector fields, Ergodic Theory Dynam. Systems, 7 (1987), 295-301.  doi: 10.1017/S0143385700004028.  Google Scholar

[22]

W. J. ZhangB. Krauskopf and V. Kirk, How to find a codimension-one heteroclinic cycle between two periodic orbits, Discrete Continuous Dynam. Systems, 32 (2012), 2825-2851.  doi: 10.3934/dcds.2012.32.2825.  Google Scholar

show all references

References:
[1]

M. A. D. AguiarS. B. S. D. Castro and I. S. Labouriau, Simple vector fields with complex behavior, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 369-381.  doi: 10.1142/S021812740601485X.  Google Scholar

[2]

V. Arnold, V. Afraimovich, Y. Ilyashenko and L. P. Shilnikov, Bifurcation Theory and Catastrophe Theory, Encyclopaedia of Mathematical Sciences, 5. V. I. Arnold (ed.), Springer-Verlag, Berlin, 1994.  Google Scholar

[3]

J. A. Beloqui, Modulus of stability for vector fields on 3-manifolds, J. Differential Equations, 65 (1986), 374-395.  doi: 10.1016/0022-0396(86)90025-2.  Google Scholar

[4]

C. Bonatti and E. Dufraine, Éuivalence topologique de connexions de selles en dimension 3, (French) [Topological equivalence of connections between 3-dimensional saddles], Ergodic Theory Dynam. Systems, 23 (2003), 1347-1381.  doi: 10.1017/S0143385703000130.  Google Scholar

[5]

M. Carvalho and A. P. Rodrigues, Complete set of invariants for a Bykov attractor, Regul. Chaotic. Dyn., 23 (2018), 227-247.  doi: 10.1134/S1560354718030012.  Google Scholar

[6]

P. ChossatM. Golubitsky and B. L. Keyfitz, Hopf-Hopf mode interactions with O(2) symmetry, Dynam. Stability Systems, 1 (1986), 255-292.  doi: 10.1080/02681118608806019.  Google Scholar

[7]

E. Dufraine, Some topological invariants for three-dimensional flows, Chaos, 11 (2011), 443-448.  doi: 10.1063/1.1385918.  Google Scholar

[8]

M. J. Field, Equivariant dynamical systems, Trans. Amer. Math. Soc., 259 (1980), 185-205.  doi: 10.1090/S0002-9947-1980-0561832-4.  Google Scholar

[9]

M. Krupa and I. Melbourne, Asymptotic stability of heteroclinic cycles in systems with symmetry, Ergodic Theory Dynam. Systems, 15 (1995), 121-147.  doi: 10.1017/S0143385700008270.  Google Scholar

[10]

M. Krupa and I. Melbourne, Asymptotic stability of heteroclinic cycles in systems with symmetry. II, Proc. Roy. Soc. Edinburg Sect. A, 134 (2004), 1177-1197.  doi: 10.1017/S0308210500003693.  Google Scholar

[11]

I. S. Labouriau and A. A. P. Rodrigues, Dense heteroclinic tangencies near a Bykov cycle, J. Differential Equations, 259 (2015), 5875-5902.  doi: 10.1016/j.jde.2015.07.017.  Google Scholar

[12]

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.  Google Scholar

[13]

I. Melbourne, Intermittency as a codimension-three phenomenon, J. Dynam. Differential Equations, 1 (1989), 347-367.  doi: 10.1007/BF01048454.  Google Scholar

[14]

J. Palis, A differentiable invariant of topological conjugacies and moduli of stability, Astérisque, 51 (1987), 335-346.   Google Scholar

[15]

A. A. P. Rodrigues, Moduli for heteroclinic connections involving Saddle-Foci and periodic solutions, Discrete Continuous Dynam. Systems, 35 (2015), 3155-3182.  doi: 10.3934/dcds.2015.35.3155.  Google Scholar

[16]

A. A. P. RodriguesI. S. Labouriau and M. A. D. Aguiar, Chaotic double cycling, Dynamical Systems: An International Journal., 26 (2011), 199-233.  doi: 10.1080/14689367.2011.557179.  Google Scholar

[17]

D. Ruelle, Historic behaviour in smooth dynamical systems, Global Analysis of Dynamical Systems, Institute of Physics Publishing, Bristol, (2001), 63–66.  Google Scholar

[18]

A. Susín and C. Simó, On moduli of conjugation for some n-dimensional vector fields, J. Differential Equations, 79 (1989), 168-177.  doi: 10.1016/0022-0396(89)90118-6.  Google Scholar

[19]

F. Takens, Partially hyperbolic fixed points, Topology, 10 (1971), 133-147.  doi: 10.1016/0040-9383(71)90035-8.  Google Scholar

[20]

F. Takens, Heteroclinic attractors: Time averages and moduli of topological conjugacy, Bol. Soc. Brasil. Mat. (N.S.), 25 (1994), 107-120.  doi: 10.1007/BF01232938.  Google Scholar

[21]

Y. Togawa, A modulus of 3-dimensional vector fields, Ergodic Theory Dynam. Systems, 7 (1987), 295-301.  doi: 10.1017/S0143385700004028.  Google Scholar

[22]

W. J. ZhangB. Krauskopf and V. Kirk, How to find a codimension-one heteroclinic cycle between two periodic orbits, Discrete Continuous Dynam. Systems, 32 (2012), 2825-2851.  doi: 10.3934/dcds.2012.32.2825.  Google Scholar

Figure 1.  Local data near a periodic solution $ {\mathcal C} $
Figure 2.  Linear components of the global maps
Figure 3.  Scheme for the global transition
Figure 4.  Phase diagram of (23)
Figure 5.  Bowen's example (24) with $ \varepsilon>0 $
Figure 6.  Illustration of the properties that are conveyed from (25) to its lifting (26)
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