Summary
A two-point boundary-value problem is considered for the differential equationx″+g(t, x, x′, λ, μ)=0 containing two real parameters. It is assumed that the problem has a solution,x 0 (t) whenλ=μ=0. Branching of solutions nearx 0 (t) is studied for small values of the parametersλ andμ. The “generic” cases of branching, in which critical quantities are nonzero, are studied. A number of cases are distinguished, which depend on the ranks of two special matrices. In each case a neighborhood of (0, 0) in the plane of the two parametersλ andμ is described. For (λ, μ) in certain subregions of this neighborhood there are several solutions of the problem which reduce tox 0 (t) as (λ, μ)→(0, 0), while for (λ, μ) in other subregions of the neighborhood there are no solutions of the problem near tox 0 (t). An application is given to periodic solutions of a damped nonlinear oscillator with equationx″+λx′+g(x)=μf(t), withλ andμ small and varying independently. Herex 0 (t) is a nonconstant periodic solution ofx″+g(x)=0.
Übersicht
Für die Differentialgleichungx″+g(t, x, x′, λ, μ)=0, die zwei reelle Parameter enthält, wird ein Zweipunkt-Grenzwert-Problem untersucht. Es wird angenommen, daß fürλ=μ=0 eine Lösungx 0 (t) vorhanden ist. Für kleine Werte der Parameterλ undμ wird die Verzweigung der Lösung in der Nähe vonx 0 (t) betrachtet. Es ergeben sich Sonderfälle für die Verzweigung, in denen kritische Größen nicht verschwinden. Abhängig von dem Rang zweier spezieller Matrizen müssen verschiedene Fälle unterschieden werden. Für jeden dieser Fälle wird die Umgebung von (0, 0) in der Ebene der Parameterλ undμ untersucht. In gewissen Bereichen dieser Umgebung gibt es mehrere Lösungen, die für (λ, μ→(0, 0) gegenx 0 (t) streben: für andere Bereiche gibt es dagegen keine Lösung in der Nähe vonx 0 (t). Als Anwendungsbeispiel werden die periodischen Lösungen eines gedämpften, nichtlinearen Schwingers mit der Gleichungx″+λx′+g(x)=μf(t) untersucht, wobeiλ undμ klein und unabhängig voneinander sind. Im vorliegenden Fall istx 0 (t) eine nichtkonstante periodische Lösung vonx″+g(x)=0.
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References
Hale, J. K.: Bifurcation with Several Parameters. VII International Conference on Nonlinear Oscillations, Berlin 1975, to appear
Hartman, P.: OnN-Parameter Families and Interpolation Problems for Nonlinear Ordinary Differential Equations. Trans. Amer. Math. Soc. 154 (1971) pp 201–226
Lasota, A.; Opial, Z.: On the Existence and Uniqueness of Solutions of a Boundary Value Problem for an Ordinary Second-Order Differential Equation. Colloq. Math. 18 (1967) pp. 1–5.
Loud, W. S.: Periodic Solutions ofx″+cx′+g(x)=εf(t). Mem. Amer. Math. Soc. 31, 1959.
Loud, W. S.: Subharmonic Solutions of Second Order Equations Arising Near Harmonic Solutions. J. Diff. Equat. 11 (1972) pp. 628–660.
Loud, W. S.: Branching of Boundary-Value Problems for Second-Order Equations. VII International Conference on Nonlinear Oscillations, Berlin, 1975, to appear.
Schmitt, K.: A Nonlinear Boundary Value Problem, J. Diff. Equat. 7 (1970) pp. 527–537
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Dedicated to Professor Karl Klotter on his seventy-fifth birthday.
The research for this paper was supported in part by Grant No. DA-ARO-D-31-124-73-G199, U.S. Army Research Office.
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Loud, W.S. Branching of solutions of two-parameter boundary-value problems for second order differential equations. Ing. arch 45, 347–359 (1976). https://doi.org/10.1007/BF02482629
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DOI: https://doi.org/10.1007/BF02482629