July  2015, 35(7): 3155-3182. doi: 10.3934/dcds.2015.35.3155

Moduli for heteroclinic connections involving saddle-foci and periodic solutions

1. 

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

Received  March 2014 Revised  December 2014 Published  January 2015

Dimension three is the lowest dimension where we can find chaotic behaviour for flows and it may be helpful to distinguish in advance ``equivalent'' complex dynamics. In this article, we give numerical invariants for the topological equivalence of vector fields on three-dimensional manifolds whose flows exhibit one-dimensional heteroclinic connections involving either saddle-foci or periodic solutions. Computed as an infinite limit time, these moduli of topological equivalence heavily rely on the behaviour near the invariant saddles. We also present an alternative proof of the Togawa's Theorem.
Citation: Alexandre A. P. Rodrigues. Moduli for heteroclinic connections involving saddle-foci and periodic solutions. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3155-3182. doi: 10.3934/dcds.2015.35.3155
References:
[1]

V. I. Arnold, V. S. Afraimovich, Y. S. Iljashenko and L. P. Shilnikov, Bifurcation Theory and Catastrophe Theory, Dynamical Systems V. Encyclopaedia of Mathematical Sciences, Springer-Verlag, New York, 1994. doi: 10.1007/978-3-642-57884-7.

[2]

M. A. D. Aguiar, S. B. Castro and I. S. Labouriau, Simple Vector Fields with Complex Behaviour, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 369-381. doi: 10.1142/S021812740601485X.

[3]

M. A. D. Aguiar, I. S. Labouriau and A. A. P. Rodrigues, Switching near a heteroclinic network of rotating nodes, Dyn. Syst., 25 (2010), 75-95. doi: 10.1080/14689360903252119.

[4]

A. Arnéodo, P. Coullet and C. Tresser, A possible new mechanism for the onset of turbulence, Phys. Lett. A, 81 (1981), 197-201. doi: 10.1016/0375-9601(81)90239-5.

[5]

A. Arnéodo, P. Coullet and C. Tresser, Possible new strange attractors with spiral structure, Comm. Math. Phys., 79 (1981), 573-579. doi: 10.1007/BF01209312.

[6]

G. R. Belitskii, Functional equations, and conjugacy of local diffeomorphisms of finite smoothness class, Funkcional. Anal. i Prilozen, 7 (1973), 17-28.

[7]

J. A. Beloqui, Módulo de Estabilidade Para Campos Vectoriais em Variedades Tridimensionais, Ph.D Thesis, IMPA-Brasil, 1981.

[8]

C. Bonatti and E. Dufraine, Equivalence topologique de connexions de selles en dimension 3, Ergodic Theory Dynam. Systems, 23 (2003), 1347-1381. doi: 10.1017/S0143385703000130.

[9]

V. V. Bykov, Orbit structure in a neighbourhood of a separatrix cycle containing two saddle-foci, Amer. Math. Soc. Transl, Serie 2, 200 (2000), 87-97.

[10]

J. C. Ceballos and R. Labarca, A note on modulus of stability for cycles of the complex type, Phys. D, 55 (1992), 37-44. doi: 10.1016/0167-2789(92)90186-Q.

[11]

B. Deng, The Shilnikov Problem, Exponential Expansion, Strong $\lambda$-Lemma, $C^1$ Linearisation and Homoclinic Bifurcation, Journal of Differential Equations, 79 (1989), 189-231. doi: 10.1016/0022-0396(89)90100-9.

[12]

B. Deng, On Shilnikov's Homoclinic Saddle-Focus Theorem, Journal of Differential Equations, 102 (1993), 305-329. doi: 10.1006/jdeq.1993.1031.

[13]

B. Deng, Exponential expansion with principal eigenvalues, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), 1161-1167. doi: 10.1142/S0218127496000655.

[14]

E. Dufraine, Some topological invariants for three-dimensional flows, Chaos, 11 (2001), 443-448. doi: 10.1063/1.1385918.

[15]

E. Dufraine, Un critére d'existence d'invariant pour la conjugaison de difféomorphismes et de champs de vecteurs, C.R. Acad. Sci. Paris, 334 (2002), 53-58. doi: 10.1016/S1631-073X(02)02207-0.

[16]

M. Field, Lectures on Bifurcations, Dynamics and Symmetry, Pitman Research Notes in Mathematics Series, 356, Longman, Harlow, 1996.

[17]

A. C. Fowler, Homoclinic bifurcations in $n$ dimensions, Stud. Appl. Math., 83 (1990), 193-209.

[18]

A. Gaunersdorfer, Time averages for heteroclinic attractors, SIAM J. Appl. Math., 52 (1992), 1476-1489. doi: 10.1137/0152085.

[19]

P. Glendinning and C. Sparrow, Local and global behavior near homoclinic orbits, J. Stat. Phys., 35 (1984), 645-696. doi: 10.1007/BF01010828.

[20]

S. V. Gonchenko, L. P. Shilnikov, O. V. Stenkin and D. V. Turaev, Bifurcations of systems with structurally unstable homoclinic orbits and moduli of $\Omega$-equivalence, Comput. Math. Appl., 34 (1997), 111-142. doi: 10.1016/S0898-1221(97)00121-1.

[21]

P. Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Math. Mexicana, 5 (1960), 220-241.

[22]

M. R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. IHES, 49 (1979), 5-233.

[23]

M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, 1974.

[24]

A. J. Homburg, Periodic attractors, strange attractors and hyperbolic dynamics near homoclinic orbit to a saddle-focus equilibria, Nonlinearity, 15 (2002), 1029-1050. doi: 10.1088/0951-7715/15/4/304.

[25]

A. J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, Handbook of Dynamical Systems, 3 (2010), 379-524. doi: 10.1016/S1874-575X(10)00316-4.

[26]

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

[27]

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

[28]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Second edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 1998.

[29]

I. S. Labouriau and A. A. P. Rodrigues, Global generic dynamics close to symmetry, Journal of Differential Equations, 253 (2012), 2527-2557. doi: 10.1016/j.jde.2012.06.009.

[30]

I. S. Labouriau and A. A. P. Rodrigues, Partial symmetry breaking and heteroclinic tangencies, in Progress and Challenges in Dynamical Systems, Springer Proc. Math. Stat., 54,, Springer-Verlag, 2013, 281-299. doi: 10.1007/978-3-642-38830-9_17.

[31]

I. M. Ovsyannikov and L. P. Shilnikov, On systems with saddle-focus homoclinic curve, Math. USSR Sbornik, 73 (1992), 415-443. doi: 10.1070/SM1987v058n02ABEH003120.

[32]

J. Palis, A differentiable invariant of topological conjugacies and moduli of stability, Dynamical systems, Vol. III, Warsaw. Soc. Math. France (Astérisque), 51 (1978), 335-346.

[33]

A. A. P. Rodrigues, Persistent switching near a heteroclinic model for the geodynamo problem, Chaos, Solitons & Fractals, 47 (2013), 73-86. doi: 10.1016/j.chaos.2012.12.005.

[34]

A. A. P. Rodrigues, Repelling dynamics near a Bykov cycle, J. Dynam. Differential Equations, 25 (2013), 605-625. doi: 10.1007/s10884-013-9289-2.

[35]

A. A. P. Rodrigues and I. S. Labouriau, Spiralling dynamics near a heteroclinic network, Phys. D, 268 (2014), 34-49. doi: 10.1016/j.physd.2013.10.012.

[36]

A. A. P. Rodrigues, I. S. Labouriau and M. A. D. Aguiar, Chaotic double cycling, Dyn. Syst., 26 (2011), 199-233. doi: 10.1080/14689367.2011.557179.

[37]

V. S. Samovol, Linearization of a system of differential equations in the neighbourhood of a singular point, Dokl. Akad. Nauk SSSR, 206 (1972), 545-548.

[38]

L. P. Shilnikov, Some cases of generation of periodic motion from singular trajectories, Math. USSR Sbornik, 61 (1963), 443-466.

[39]

L. P. Shilnikov, A case of the existence of a denumerable set of periodic motions, Sov. Math. Dokl., 160 (1965), 558-561.

[40]

L. P. Shilnikov, The existence of a denumerable set of periodic motions in four dimensional space in an extended neighbourhood of a saddle-focus, Sov. Math. Dokl., 172 (1967), 54-57.

[41]

L. Shilnikov, A. Shilnikov, D. Turaev and L. Chua, Methods of Qualitative Theory in Nonlinear Dynamics I, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. doi: 10.1142/9789812798596.

[42]

L. Shilnikov, A. Shilnikov, D. Turaev and L. Chua, Methods of Qualitative Theory in Nonlinear Dynamics II, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812798558.

[43]

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

[44]

Y. Togawa, A modulus of $3$-dimensional vector fields, Ergod. Theory Dyn. Syst., 7 (1987), 295-301. doi: 10.1017/S0143385700004028.

[45]

C. Tresser, About some theorems by L. P. Shilnikov, Ann. Inst. H. Poincaré, 40 (1984), 441-461.

[46]

S. J. van Strien, One Parameter Families of Vector Fields, Bifurcations near Saddle-Connections, Ph.D. Thesis, Utrecht University, 1982.

[47]

S. Wiggins, Introduction in Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, TAM 2, New York, 1990. doi: 10.1007/978-1-4757-4067-7.

show all references

References:
[1]

V. I. Arnold, V. S. Afraimovich, Y. S. Iljashenko and L. P. Shilnikov, Bifurcation Theory and Catastrophe Theory, Dynamical Systems V. Encyclopaedia of Mathematical Sciences, Springer-Verlag, New York, 1994. doi: 10.1007/978-3-642-57884-7.

[2]

M. A. D. Aguiar, S. B. Castro and I. S. Labouriau, Simple Vector Fields with Complex Behaviour, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 369-381. doi: 10.1142/S021812740601485X.

[3]

M. A. D. Aguiar, I. S. Labouriau and A. A. P. Rodrigues, Switching near a heteroclinic network of rotating nodes, Dyn. Syst., 25 (2010), 75-95. doi: 10.1080/14689360903252119.

[4]

A. Arnéodo, P. Coullet and C. Tresser, A possible new mechanism for the onset of turbulence, Phys. Lett. A, 81 (1981), 197-201. doi: 10.1016/0375-9601(81)90239-5.

[5]

A. Arnéodo, P. Coullet and C. Tresser, Possible new strange attractors with spiral structure, Comm. Math. Phys., 79 (1981), 573-579. doi: 10.1007/BF01209312.

[6]

G. R. Belitskii, Functional equations, and conjugacy of local diffeomorphisms of finite smoothness class, Funkcional. Anal. i Prilozen, 7 (1973), 17-28.

[7]

J. A. Beloqui, Módulo de Estabilidade Para Campos Vectoriais em Variedades Tridimensionais, Ph.D Thesis, IMPA-Brasil, 1981.

[8]

C. Bonatti and E. Dufraine, Equivalence topologique de connexions de selles en dimension 3, Ergodic Theory Dynam. Systems, 23 (2003), 1347-1381. doi: 10.1017/S0143385703000130.

[9]

V. V. Bykov, Orbit structure in a neighbourhood of a separatrix cycle containing two saddle-foci, Amer. Math. Soc. Transl, Serie 2, 200 (2000), 87-97.

[10]

J. C. Ceballos and R. Labarca, A note on modulus of stability for cycles of the complex type, Phys. D, 55 (1992), 37-44. doi: 10.1016/0167-2789(92)90186-Q.

[11]

B. Deng, The Shilnikov Problem, Exponential Expansion, Strong $\lambda$-Lemma, $C^1$ Linearisation and Homoclinic Bifurcation, Journal of Differential Equations, 79 (1989), 189-231. doi: 10.1016/0022-0396(89)90100-9.

[12]

B. Deng, On Shilnikov's Homoclinic Saddle-Focus Theorem, Journal of Differential Equations, 102 (1993), 305-329. doi: 10.1006/jdeq.1993.1031.

[13]

B. Deng, Exponential expansion with principal eigenvalues, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), 1161-1167. doi: 10.1142/S0218127496000655.

[14]

E. Dufraine, Some topological invariants for three-dimensional flows, Chaos, 11 (2001), 443-448. doi: 10.1063/1.1385918.

[15]

E. Dufraine, Un critére d'existence d'invariant pour la conjugaison de difféomorphismes et de champs de vecteurs, C.R. Acad. Sci. Paris, 334 (2002), 53-58. doi: 10.1016/S1631-073X(02)02207-0.

[16]

M. Field, Lectures on Bifurcations, Dynamics and Symmetry, Pitman Research Notes in Mathematics Series, 356, Longman, Harlow, 1996.

[17]

A. C. Fowler, Homoclinic bifurcations in $n$ dimensions, Stud. Appl. Math., 83 (1990), 193-209.

[18]

A. Gaunersdorfer, Time averages for heteroclinic attractors, SIAM J. Appl. Math., 52 (1992), 1476-1489. doi: 10.1137/0152085.

[19]

P. Glendinning and C. Sparrow, Local and global behavior near homoclinic orbits, J. Stat. Phys., 35 (1984), 645-696. doi: 10.1007/BF01010828.

[20]

S. V. Gonchenko, L. P. Shilnikov, O. V. Stenkin and D. V. Turaev, Bifurcations of systems with structurally unstable homoclinic orbits and moduli of $\Omega$-equivalence, Comput. Math. Appl., 34 (1997), 111-142. doi: 10.1016/S0898-1221(97)00121-1.

[21]

P. Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Math. Mexicana, 5 (1960), 220-241.

[22]

M. R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. IHES, 49 (1979), 5-233.

[23]

M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, 1974.

[24]

A. J. Homburg, Periodic attractors, strange attractors and hyperbolic dynamics near homoclinic orbit to a saddle-focus equilibria, Nonlinearity, 15 (2002), 1029-1050. doi: 10.1088/0951-7715/15/4/304.

[25]

A. J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, Handbook of Dynamical Systems, 3 (2010), 379-524. doi: 10.1016/S1874-575X(10)00316-4.

[26]

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

[27]

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

[28]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Second edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 1998.

[29]

I. S. Labouriau and A. A. P. Rodrigues, Global generic dynamics close to symmetry, Journal of Differential Equations, 253 (2012), 2527-2557. doi: 10.1016/j.jde.2012.06.009.

[30]

I. S. Labouriau and A. A. P. Rodrigues, Partial symmetry breaking and heteroclinic tangencies, in Progress and Challenges in Dynamical Systems, Springer Proc. Math. Stat., 54,, Springer-Verlag, 2013, 281-299. doi: 10.1007/978-3-642-38830-9_17.

[31]

I. M. Ovsyannikov and L. P. Shilnikov, On systems with saddle-focus homoclinic curve, Math. USSR Sbornik, 73 (1992), 415-443. doi: 10.1070/SM1987v058n02ABEH003120.

[32]

J. Palis, A differentiable invariant of topological conjugacies and moduli of stability, Dynamical systems, Vol. III, Warsaw. Soc. Math. France (Astérisque), 51 (1978), 335-346.

[33]

A. A. P. Rodrigues, Persistent switching near a heteroclinic model for the geodynamo problem, Chaos, Solitons & Fractals, 47 (2013), 73-86. doi: 10.1016/j.chaos.2012.12.005.

[34]

A. A. P. Rodrigues, Repelling dynamics near a Bykov cycle, J. Dynam. Differential Equations, 25 (2013), 605-625. doi: 10.1007/s10884-013-9289-2.

[35]

A. A. P. Rodrigues and I. S. Labouriau, Spiralling dynamics near a heteroclinic network, Phys. D, 268 (2014), 34-49. doi: 10.1016/j.physd.2013.10.012.

[36]

A. A. P. Rodrigues, I. S. Labouriau and M. A. D. Aguiar, Chaotic double cycling, Dyn. Syst., 26 (2011), 199-233. doi: 10.1080/14689367.2011.557179.

[37]

V. S. Samovol, Linearization of a system of differential equations in the neighbourhood of a singular point, Dokl. Akad. Nauk SSSR, 206 (1972), 545-548.

[38]

L. P. Shilnikov, Some cases of generation of periodic motion from singular trajectories, Math. USSR Sbornik, 61 (1963), 443-466.

[39]

L. P. Shilnikov, A case of the existence of a denumerable set of periodic motions, Sov. Math. Dokl., 160 (1965), 558-561.

[40]

L. P. Shilnikov, The existence of a denumerable set of periodic motions in four dimensional space in an extended neighbourhood of a saddle-focus, Sov. Math. Dokl., 172 (1967), 54-57.

[41]

L. Shilnikov, A. Shilnikov, D. Turaev and L. Chua, Methods of Qualitative Theory in Nonlinear Dynamics I, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. doi: 10.1142/9789812798596.

[42]

L. Shilnikov, A. Shilnikov, D. Turaev and L. Chua, Methods of Qualitative Theory in Nonlinear Dynamics II, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812798558.

[43]

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

[44]

Y. Togawa, A modulus of $3$-dimensional vector fields, Ergod. Theory Dyn. Syst., 7 (1987), 295-301. doi: 10.1017/S0143385700004028.

[45]

C. Tresser, About some theorems by L. P. Shilnikov, Ann. Inst. H. Poincaré, 40 (1984), 441-461.

[46]

S. J. van Strien, One Parameter Families of Vector Fields, Bifurcations near Saddle-Connections, Ph.D. Thesis, Utrecht University, 1982.

[47]

S. Wiggins, Introduction in Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, TAM 2, New York, 1990. doi: 10.1007/978-1-4757-4067-7.

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