Summary
In this note we use a new averaging method, which was introduced in [2], to explain the geometrical behaviour of systems governed by nonlinear boundary value problems of the formy″+g(y)=K sin(Ωt),y(0)=y(π/Ω)=0. We show by numerical computations that global features of the solutions (such as the number of solutions, their magnitude, bifurcation behaviour, etc.) agree in both the original and averaged model. As an example, the pendulum equation is discussed in detail.
Zusammenfassung
In dieser Arbeit benutzen wir eine neue, in [2] eingeführte Mittelwertmethode, um das geometrische Verhalten nichtlinearer Randwertprobleme der Formy″+g(y)=K sin(Ωt),y(0)=y(π/Ω)=0. zu erklären. Wir belegen durch numerische Untersuchungen, daß globale Eigenschaften der Lösungen (wie z. B. die Anzahl der Lösungen, ihre Größenordnung, das Verzweigungsverhalten usw.) in der originalen und genäherten Gleichung übereinstimmen. Als Beispiel wird die Pendelgleichung ausführlich diskutiert.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Amman,Existence of multiple solutions for nonlinear elliptic boundary value problems. Indiana Univ. Math. J.21, 925–935 (1972).
N. Bazley and L. Brüll,Periodic solutions of an averaged Duffing equation, Acta Polyt. Scand., ME No. 90 (1985).
N. Bazley and R. Weinacht,A class of operator equations in nonlinear mechanics. Applicable Analysis15, 7–14 (1983).
H. Berestycki and P. C. Lions,Nonlinear scalar field equations I, Existence of a ground state. Arch. Rat. Mech. Anal.82, 313–345 (1983).
S. Bernfeld and V. Lakshmikantham,An Introduction to Nonlinear Boundary Value Problems. Academic Press, New York 1974.
G. Birkhoff and G.-C. Rota,Ordinary Differential Equations. Wiley, New York 1969.
M. Braun,Differential Equations and Their Applications. Springer, New York 1975.
L. Brüll,On the number of subharmonic solutions of the forced pendulum equation, Report 82-10. Applied Mathematics, University of Cologne.
L. Brüll and H. Lange,Stationary, oscillatory and solitary wave type solution of singular nonlinear Schrödinger equations. Math. Meth. Appl. Sc., to appear appr. Vol. 8.
L. Brüll and J. Mawhin,Finiteness of the set of solutions of some boundary-value problems for ordinary differential equations, to appear.
S.-N. Chow and J. Hale,Methods of Bifurcation Theory. Springer, New York 1982.
G. Duffing,Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung. Vieweg & Sohn, Braunschweig 1918.
H. Ehrmann,Über die Existenz der Lösungen von Randwertaufgaben bei gewöhnlichen Differentialgleichungen zweiter Ordnung. Math. Ann.134, 167–194 (1957).
J. Guckenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Springer, New York 1983.
P. Hagedorn,Nichtlineare Schwingungen. Akademische Verlagsges., Wiesbaden 1978.
J. Hale,Ordinary Differential Equations. Wiley, New York 1969.
B. Huberman, J. Crutchfield and N. Packard,Noise phenomena in Josephson junctions. Appl. Phys. Lett.37, 8, 750–752 (1980).
H. Moriguchi and T. Nakamura,Forced oscillations of systems with nonlinear restoring force. J. Phys. Soc. Japan52, 3, 732–743 (1983).
A. Nayfeh and D. Mook,Nonlinear Oscillations. Wiley, New York 1979.
E. Sanders, Diplomarbeit, Universität zu Köln (1985).
H. Steins, Diplomarbeit, Universität zu Köln (1984).
J. Stoker,Nonlinear Vibrations in Mechanical and Electrical Systems. Intersciences Publ., New York 1950.
Author information
Authors and Affiliations
Additional information
Supported by the Deutsche Forschungsgemeinschaft under grant No. BA 735/3-1
Rights and permissions
About this article
Cite this article
Brüll, L., Hölters, H.P. A geometrical approach to bifurcation for nonlinear boundary value problems. Journal of Applied Mathematics and Physics (ZAMP) 37, 820–830 (1986). https://doi.org/10.1007/BF00953673
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00953673