Gonchenko, S. V.; Shil’nikov, L. P. Invariants of \(\Omega\)-conjugacy of diffeomorphisms with a nongeneric homoclinic trajectory. (English. Russian original) Zbl 0705.58044 Ukr. Math. J. 42, No. 2, 134-140 (1990); translation from Ukr. Mat. Zh. 42, No. 2, 153-159 (1990). We consider a smooth \(C^ r\) (r\(\geq 3)\) diffeomorphism T of a 2- dimensional manifold which satisfies the condition: 1) T has a saddle fixpoint with eigenvalues \(\lambda\), \(\gamma\), \(0<| \lambda | <1<| \gamma |\); 2) the saddle value \(\sigma =| \lambda \gamma | <1\); 3) T has a nonhyperbolic homoclinic orbit which consists of the stable and unstable manifolds \(W^ s\), \(W^ u\) of a saddle of odd order \(n\geq 1.\) The authors have obtained necessary conditions for the topological conjugate of a homoclinic orbit [see Ukr. Math. J. 39, 15-21 (1987); translation from Ukr. Mat. Zh. 39, No.1, 21-28 (1987; Zbl 0635.58025)]. The author gives sufficient conditions for \(\Omega\)-conjugacy as follows: Let \[ \theta =-\ln | \lambda | /\ln | \gamma |,\quad \tau =(1/\ln | \gamma |)\ln | cx^+/y^-|; \] and let \((M^+(x^+,0),M^-(0,y^-))\) be a pair of points in \(\Gamma\). If \(\theta_ 1=\theta_ 2=p/g\), \(s/g<\tau^ k_ 0<(s+1)/g\), \(k=1,2\), for some integer s, then the diffeomorphisms \(T_ 1\) and \(T_ 2\) are \(\Omega\)-conjugate in some neighbourhood of the homoclinic orbit \(\Gamma\), where \(\tau_ 0\) is defined by the formula \[ \tau (T^ sM^+,T^ pM^-)=\tau (M^+,M^-)+(s-p)(1-\theta). \] Reviewer: Sheng Liren Cited in 3 ReviewsCited in 25 Documents MSC: 37C80 Symmetries, equivariant dynamical systems (MSC2010) 54H20 Topological dynamics (MSC2010) 37B99 Topological dynamics 57R50 Differential topological aspects of diffeomorphisms Keywords:dynamical systems; stable manifold; unstable manifold; saddle value; homoclinic orbit; conjugacy Citations:Zbl 0635.58025 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] S. V. Gonchenko and L. P. Shil’nikov, ?Arithmetic properties of topological invariants of systems with a nongeneric homoclinic trajectory,? Ukr. Mat. Zh.,39, No. 1, 21-28 (1987). · Zbl 0635.58025 · doi:10.1007/BF01056417 [2] A. Kelley, ?The stable, center-stable, center, center-unstable and unstable manifolds,? J. Diff. Equat.,3, No. 4, 546-570 (1967). · Zbl 0173.11001 · doi:10.1016/0022-0396(67)90016-2 [3] M. W. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lect. Notes in Math., Springer Verlag, Berlin (1983). [4] I. M. Ovsyannikov and L. P. Shil’nikov, ?Systems with a homoclinic curve of saddlefocus type,? Mat. Sb.,130, No. 4, 552-570 (1986). [5] N. K. Gavrilov and L. P. Shil’nikov, ?Three-dimensional dynamical systems close to a system with a nongeneric homoclinic trajectory, I, II,? Mat. Sb.,88, No. 4, 475-492 (1972);90, No. 1, 139-157 (1973). [6] S. V. Gonchenko and L. P. Shil’nikov, ?Dynamical systems with nongeneric homoclinic trajectories,? Dokl. Akad. Nauk SSSR,292, No. 5, 1049-1053 (1985). [7] V. S. Afraimovich and L. P. Shil’nikov, ?Distinguished sets of Morse-Smale systems,? Tr. Mosk. Mat. Obshch.,28, 181-214 (1973). [8] S. V. Gonchenko, ?Nontrivial hyperbolic subsets of systems with a nongeneric homoclinic curve,? in: Methods of Qualitative Theory of Differential Equations [in Russian], Gor’kii (1984), pp. 89-102. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.