zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Bifurcation from a periodic orbit in perturbed planar Hamiltonian systems. (English) Zbl 1039.70012
The authors study the perturbed plane differential system $u'= -J\nabla H(u)+ p(\varepsilon, t,u)$. Here $t$ is a variable, $p$ is Carathéodory function, $T$ is periodic in variable $t$, and $j$ is a symplectic matrix. $T$-periodic functions are looked for. Theorems are proved which give conditions for the existence of $T$-periodic solution. Some properties of the time map are discussed, and an application of periodic solutions close to the homoclinic ones is studied.

MSC:
70H09Perturbation theories (mechanics of particles and systems)
70K44Homoclinic and heteroclinic trajectories (nonlinear dynamics)
70K50Transition to stochasticity (general mechanics)
WorldCat.org
Full Text: DOI
References:
[1] Alfawicka, B.: Inverse problems connected with periods of oscillations described x \ddot{}+$g(x)=0$. Ann. polon. Math. 44, 297-308 (1984) · Zbl 0561.34027
[2] Ambrosetti, A.; Badiale, M.: Homoclinics: Poincaré--Melnikov type results via a variational approach. Ann. inst. H. Poincaré anal. Non linéaire 15, 127-149 (1998)
[3] Ambrosetti, A.; Badiale, M.: Variational perturbative methods and bifurcation of bound states from the essential spectrum. Proc. roy. Soc. Edinburgh sect. A 128, 1131-1161 (1998) · Zbl 0928.34029
[4] Ambrosetti, A.; Zelati, V. Coti: Periodic solutions of singular systems. Progress in nonlinear differential equations and their applications 10 (1993) · Zbl 0785.34032
[5] Ambrosetti, A.; Zelati, V. Coti; Ekeland, I.: Symmetry breaking in Hamiltonian systems. J. differential equations 67, 165-184 (1987) · Zbl 0606.58043
[6] Berti, M.; Bolle, P.: Homoclinics and chaotic behaviour for perturbed second order systems. Ann. mat. Pura appl. (4) 176, 323-378 (1999) · Zbl 0957.37019
[7] Battelli, F.; Palmer, K. J.: Chaos in the Duffing equation. J. differential equations 101, 276-301 (1993) · Zbl 0772.34040
[8] Blaine, L. G.: Subharmonic solutions of periodically forced predator-prey systems. Nonlinear anal. 22, 739-752 (1994) · Zbl 0810.92019
[9] Buttazzoni, P.; Fonda, A.: Periodic perturbations of scalar second order differential equations. Discrete contin. Dynam. systems 3, 451-455 (1997) · Zbl 0951.34024
[10] Capietto, A.; Mawhin, J.; Zanolin, F.: Continuation theorems for periodic perturbations of autonomous systems. Trans. amer. Math. soc. 329, 41-72 (1992) · Zbl 0748.34025
[11] Chicone, C.: Bifurcation of nonlinear oscillations and frequency entrainment near resonance. SIAM J. Math. anal. 23, 1577-1608 (1992) · Zbl 0765.58018
[12] Chouikha, R.; Cuvelier, F.: Remarks on some monotonicity conditions for the period function. Appl. math. 26, 243-252 (1999) · Zbl 0993.37029
[13] Chow, S. N.; Hale, J. K.: Methods of bifurcation theory. Grundlehren der mathematischen wissenshaften 251 (1982)
[14] Chow, S. N.; Hale, J. K.; Mallet-Paret, J.: An example of bifurcation to homoclinic orbits. J. differential equations 37, 351-373 (1980) · Zbl 0439.34035
[15] Conley, C.: Isolated invariant sets and the Morse index. CBMS regional conference series in mathematics 38 (1978) · Zbl 0397.34056
[16] Felmer, P.; Manásevich, R.: A global approach for bifurcation from a nondegenerate periodic solution. Nonlinear anal. 22, 353-361 (1994) · Zbl 0807.34053
[17] Gorenflo, R.; Vessella, S.: Abel integral equations. Analysis and applications. Lecture notes in mathematics 1461 (1991) · Zbl 0717.45002
[18] Greenberg, M. J.: Lectures on algebraic topology. (1973) · Zbl 0278.35074
[19] Guckenheimer, J.; Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Applied mathematical sciences 42 (1983) · Zbl 0515.34001
[20] Hale, J. K.; Táboas, P.: Interaction of damping and forcing in a second order equation. Nonlinear anal. 2, 77-84 (1978) · Zbl 0369.34014
[21] Hale, J. K.; Táboas, P.: Bifurcation near degenerate families. Appl. anal. 11, 21-37 (1980) · Zbl 0441.34033
[22] Han, M.: Bifurcation theory of invariant tori of planar periodic perturbed systems. Sci. China ser. A 39, 509-519 (1996) · Zbl 0864.34028
[23] Hausrath, A. R.; Manásevich, R. F.: The characterization of degenerate and nondegenerate systems. Rocky mountain J. Math. 16, 203-214 (1986) · Zbl 0591.34038
[24] Hausrath, A. R.; Manásevich, R. F.: Periodic solutions of periodically forced nondegenerate systems. Rocky mountain J. Math. 18, 49-65 (1988) · Zbl 0659.34040
[25] M. Henrard, Topological degree methods in boundary value problems: existence and multiplicity results for second order differential equations, Ph.D. thesis, Université catholique de Louvain (1995) · Zbl 0831.34026
[26] Henrard, M.: Homoclinic and multibump solutions for perturbed second order systems using topological degree. Discrete contin. Dynam. systems 5, 765-782 (1999) · Zbl 0980.34043
[27] Keller, J. B.: Inverse problems. Amer. math. Monthly 83, 107-118 (1976)
[28] Krasnosels’kiı\breve , M. A.; Zabreı\breve ko, P. P.: Geometrical methods of nonlinear analysis. (1984)
[29] Kuratowski, K.: Topology, vol. 2. (1968)
[30] Kurzweil, J.: Ordinary differential equations. Studies in applied mechanics 13 (1986) · Zbl 0667.34002
[31] Landau, L. D.; Lifshitz, E. M.: Mechanics. Course of mathematical physics 1 (1976)
[32] Lazer, A. C.: Small periodic perturbations of a class of conservative systems. J. differential equations 13, 438-456 (1973) · Zbl 0287.34039
[33] Leray, J.; Schauder, J.: Topologie et équations fonctionnelle. Ann. sci. École norm. Sup. 51, 45-78 (1934) · Zbl 60.0322.02
[34] Loud, W. S.: Periodic solutions of x”+cx’+$g(x)={\epsilon}f(t)$. Mem. amer. Math. soc. 31 (1959) · Zbl 0085.30701
[35] Luo, D.; Wang, X.; Zhu, D.; Han, M.: Bifurcation theory and methods of dynamical systems. (1997) · Zbl 0961.37015
[36] Mawhin, J.: Topological degree methods in nonlinear boundary value problems. CBMS regional conference series in mathematics 40 (1979) · Zbl 0414.34025
[37] Mawhin, J.: Leray--Schauder continuation theorems in the absence of a priori bounds. Topol. methods nonlinear anal. 9, 179-200 (1997) · Zbl 0912.47040
[38] Mawhin, J.; Willem, M.: Critical point theory and Hamiltonian systems. Applied mathematical sciences 74 (1989) · Zbl 0676.58017
[39] Mazzi, L.; Sabatini, M.: A characterization of centres via first integrals. J. differential equations 76, 222-237 (1988) · Zbl 0667.34036
[40] Miranda, C.: Problemi di esistenza in analisis funzionale. Scuola normale superiore, classe di scienze, a.a. 1948--49 (1975)
[41] Obi, C.: Analytical theory of nonlinear oscillations. VII. the periods of the periodic solutions of the equation x \ddot{}+$g(x)=0$. J. math. Anal. appl. 55, 295-301 (1976) · Zbl 0359.34038
[42] Opial, Z.: Sur LES périodes des solutions de l’équation différentielle x”+$g(x)=0$. Ann. polon. Math. 10, 49-72 (1961) · Zbl 0096.29604
[43] Palmer, K. J.: Exponential dichotomies and transversal homoclinic points. J. differential equations 55, 225-256 (1984) · Zbl 0508.58035
[44] Poincaré, H.: Sur certaines solutions particulières du problème des trois corps. C. R. Acad. sci. Paris 97, 251-252 (1883) · Zbl 15.0833.01
[45] Poincaré, H.: Sur certaines solutions particulières du problème des trois corps. Bull. astron. 1, 65-75 (1884) · Zbl 15.0833.01
[46] Rabinowitz, P.: Some global results for nonlinear eigenvalue problems. J. funct. Anal. 7, 487-513 (1971) · Zbl 0212.16504
[47] Rebelo, C.: A note on uniqueness of Cauchy problems to planar Hamiltonian systems. Portugal math. 57, 415-419 (2000) · Zbl 0979.37032
[48] Reeken, M.: Stability of critical points under small perturbations. II. analytic theory. Manuscripta math. 8, 69-92 (1973) · Zbl 0248.58004
[49] Rothe, F.: The periods of the Volterra--Lotka system. J. reine angew. Math. 355, 129-138 (1985) · Zbl 0547.92011
[50] Rothe, F.: The energy-period function and perturbations of Hamiltonian systems in the plane. Canad. math. Soc. conference Proceedings 8, 621-635 (1987) · Zbl 0648.34049
[51] Rouche, N.; Mawhin, J.: Equations différentielles ordinaires. (1973) · Zbl 0289.34001
[52] Schaaf, R.: A class of Hamiltonian systems with increasing periods. J. reine angew. Math. 363, 96-109 (1985) · Zbl 0565.34037
[53] Schaaf, R.: Global solution branches of two point boundary value problems. Lecture notes in mathematics 1458 (1990) · Zbl 0780.34010
[54] Táboas, P.: Periodic solutions of a forced Lotka--Volterra equation. J. math. Anal. appl. 124, 82-97 (1987) · Zbl 0644.34032
[55] Urabe, M.: Nonlinear autonomous oscillations. Analytical theory. (1967) · Zbl 0154.09503
[56] Vanderbauwhede, R.: Local bifurcation and symmetry. Research notes in mathematics (1982) · Zbl 0539.58022
[57] Waldvogel, J.: The period in the Lotka--Volterra system is monotonic. J. math. Anal. appl. 114, 178-184 (1986) · Zbl 0588.92018
[58] Willem, M.: Perturbations of nondegenerate periodic orbits of Hamiltonian systems. NATO adv. Sci. inst. Ser. C: math. Phys. sci. 209, 261-265 (1987) · Zbl 0667.58059
[59] Zeidler, E.: Functional analysis and its applications I: Fixed-point theorems. (1986) · Zbl 0583.47050