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 & Continuous Dynamical Systems - A, 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, (1994).  doi: 10.1007/978-3-642-57884-7.  Google Scholar

[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.  doi: 10.1142/S021812740601485X.  Google Scholar

[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.  doi: 10.1080/14689360903252119.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

J. A. Beloqui, Módulo de Estabilidade Para Campos Vectoriais em Variedades Tridimensionais,, Ph.D Thesis, (1981).   Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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.  doi: 10.1016/S1631-073X(02)02207-0.  Google Scholar

[16]

M. Field, Lectures on Bifurcations, Dynamics and Symmetry,, Pitman Research Notes in Mathematics Series, (1996).   Google Scholar

[17]

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

[18]

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

[19]

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

[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.  doi: 10.1016/S0898-1221(97)00121-1.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[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.  doi: 10.1017/S0308210500003693.  Google Scholar

[28]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Second edition,, Applied Mathematical Sciences, (1998).   Google Scholar

[29]

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

[30]

I. S. Labouriau and A. A. P. Rodrigues, Partial symmetry breaking and heteroclinic tangencies,, in Progress and Challenges in Dynamical Systems, (2013), 281.  doi: 10.1007/978-3-642-38830-9_17.  Google Scholar

[31]

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

[32]

J. Palis, A differentiable invariant of topological conjugacies and moduli of stability,, Dynamical systems, 51 (1978), 335.   Google Scholar

[33]

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

[34]

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

[35]

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

[36]

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

[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.   Google Scholar

[38]

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

[39]

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

[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.   Google Scholar

[41]

L. Shilnikov, A. Shilnikov, D. Turaev and L. Chua, Methods of Qualitative Theory in Nonlinear Dynamics I,, World Scientific Publishing Co., (1998).  doi: 10.1142/9789812798596.  Google Scholar

[42]

L. Shilnikov, A. Shilnikov, D. Turaev and L. Chua, Methods of Qualitative Theory in Nonlinear Dynamics II,, World Scientific Publishing Co., (2001).  doi: 10.1142/9789812798558.  Google Scholar

[43]

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

[44]

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

[45]

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

[46]

S. J. van Strien, One Parameter Families of Vector Fields, Bifurcations near Saddle-Connections,, Ph.D. Thesis, (1982).   Google Scholar

[47]

S. Wiggins, Introduction in Applied Nonlinear Dynamical Systems and Chaos,, Springer-Verlag, (1990).  doi: 10.1007/978-1-4757-4067-7.  Google Scholar

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, (1994).  doi: 10.1007/978-3-642-57884-7.  Google Scholar

[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.  doi: 10.1142/S021812740601485X.  Google Scholar

[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.  doi: 10.1080/14689360903252119.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

J. A. Beloqui, Módulo de Estabilidade Para Campos Vectoriais em Variedades Tridimensionais,, Ph.D Thesis, (1981).   Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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.  doi: 10.1016/S1631-073X(02)02207-0.  Google Scholar

[16]

M. Field, Lectures on Bifurcations, Dynamics and Symmetry,, Pitman Research Notes in Mathematics Series, (1996).   Google Scholar

[17]

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

[18]

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

[19]

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

[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.  doi: 10.1016/S0898-1221(97)00121-1.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[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.  doi: 10.1017/S0308210500003693.  Google Scholar

[28]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Second edition,, Applied Mathematical Sciences, (1998).   Google Scholar

[29]

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

[30]

I. S. Labouriau and A. A. P. Rodrigues, Partial symmetry breaking and heteroclinic tangencies,, in Progress and Challenges in Dynamical Systems, (2013), 281.  doi: 10.1007/978-3-642-38830-9_17.  Google Scholar

[31]

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

[32]

J. Palis, A differentiable invariant of topological conjugacies and moduli of stability,, Dynamical systems, 51 (1978), 335.   Google Scholar

[33]

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

[34]

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

[35]

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

[36]

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

[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.   Google Scholar

[38]

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

[39]

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

[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.   Google Scholar

[41]

L. Shilnikov, A. Shilnikov, D. Turaev and L. Chua, Methods of Qualitative Theory in Nonlinear Dynamics I,, World Scientific Publishing Co., (1998).  doi: 10.1142/9789812798596.  Google Scholar

[42]

L. Shilnikov, A. Shilnikov, D. Turaev and L. Chua, Methods of Qualitative Theory in Nonlinear Dynamics II,, World Scientific Publishing Co., (2001).  doi: 10.1142/9789812798558.  Google Scholar

[43]

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

[44]

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

[45]

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

[46]

S. J. van Strien, One Parameter Families of Vector Fields, Bifurcations near Saddle-Connections,, Ph.D. Thesis, (1982).   Google Scholar

[47]

S. Wiggins, Introduction in Applied Nonlinear Dynamical Systems and Chaos,, Springer-Verlag, (1990).  doi: 10.1007/978-1-4757-4067-7.  Google Scholar

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