Abstract
This chapter is devoted to presenting a panorama of the problems and methods of the theory of stability for ordinary differential equations. Roughly speaking, one could say that stability is the continuous dependence of the solutions as functions of the data on infinite intervals; the reason it is necessary to start a new chapter is that the theorems on continuous dependence in Chapter III are only valid on compact intervals. Keeping in mind that in general, physical phenomena develop over infinite intervals of time, the following observations may convince the reader of the necessity of undertaking a study of stability.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Z. Artstein, Uniform asymptotic stability via the limiting equation, J. Diff. Eq., 27 (1978), 172–189.
I. Babuska, M. Prager and V. Vitasek, Numerical Processes in Differential Equations, Interscience, New York, 1966.
S. Bernfeld and V. Lakshmikantham, Differential inequalities and the Okamura function, Ann. Mat. Pura Appl., 96 (1973), 89–105.
F. Brauer, Some Stability and Perturbation Problems for Differential and Integral Equations, IMPA, Rio de Janeiro, 1976.
T. A. Burton, Differential inequalities for Liapunov functions, Nonlinear Anal., Theory, Methods, Appl., 1 (1977), 331–338.
L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Springer-Verlag, Berlin, 1963.
S. N. Chow and J. A. Yorke, Lyapunov Theory and perturbation of stable and asymptotically stable systems, J. Diff. Eq., 15 (1974), 308–321.
L. Collatz, Functional Analysis and Numerical Mathematics, Academic Press, New York, 1966.
L. Collatz, The Numerical Treatment of Differential Equations, Springer-Verlag, Berlin, 1966.
W, A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Heath, Boston, 1965.
W. A. Coppel, Dichotomies in Stability Theory, Springer-Verlag, Lecture Notes in Math., vol. 629, 1978.
G. R. Gavalas, Nonlinear Differential Equations of Chemically Reacting Systems, Springer-Verlag, Berlin, 1968.
C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1971.
W. Hahn, Stability of Motion, Springer-Verlag, Berlin, 1967.
A. Halanay, Differential Equations: Stability, Oscillations, Time Lag, Academic Press, New York, 1966.
J. K. Hale and J. P. LaSalle, Differential equations: linearity versus nonlinearity, SIAM Rev., 5 (1963), 249–272.
P. Hartman and C. Olech, On global asymptotic stability of solutions of differential equations, Trans. Amer. Math. Soc., 104 (1962), 154–178.
F. John, Lectures on Advanced Numerical Analysis, Nelson, London, 1966.
V. Jurdjevic, and J. P. Quinn, Controllability and Stability, J. Diff. Eq., 25 (1978), 281–289.
V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, vol. I, Academic Press, New York, 1969.
J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, 1976.
S. Lefschetz, Differential Equations: Geometric Theory, Interscience, New York, 1957.
P. Marocco, A study of asymptotic behavior and stability of the solutions of Volterra integral equations using the topological degree, J. Diff. Eq. 43 (1982), 235–248.
C. Olech, On the global stability of an autonomous system on the plane, Contrib. Diff. Eq., vol. I, (1963), 389–400.
C. Olech, Global phase-portrait of a plane autonomous system, Ann. Inst. Fourier 14, 1 (1964), 87–98.
J. M. Ortega, Numerical Analysis: A Second Course, Academic Press, New York, 1972.
J. M. Ortega and W. C. RheinItalict, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
V. M. Popov, Hyperstability of Control Systems, Springer-Verlag, Berlin, 1973.
R. Reissig, G. Sansone, and R. Conti, Non-linear Differential Equations of Higher Order, Nordhoff, Leyden, 1974.
L. F. Shampine and M. K. Gordon, Computer Solution of Ordinary Differential Equations (The Initial Value Problem), Freeman, San Francisco, 1975.
H. J. Stetter, Analysis of Discretization Methods for Ordinary Differential Equations, Springer-Verlag, Berlin, 1973.
J. J. Stoker, On the stability of mechanical systems, Comm. Pure Appl. Math., 5 (1955), 133–142.
A. H. Stroud, Numerical Quadrature and Solution of Ordinary Differential Equations, Springer-Verlag, Berlin, 1974.
J. Todd, Survey of Numerical Analysis, McGraw-Hill, New York, 1962.
G. Vidossich, Two remarks on the stability of ordinary differential equations, Nonlinear Anal., TMA, 4 (1980), 967–974.
J. A. Yorke, A theorem on Liapunov functions using V, Math. System T., 4 (1969), 40–45.
J. A. Yorke, Differential inequalities and non-Lipschitz scalar functions, Math. System T., 4 (1969), 140–153.
T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Springer-Verlag, Berlin, New York, 1975.
S. R. Bernfeld and L. Salvadori, Generalized Hopf bifurcation and h-asymptotic stability, Nonlin. Anal. TMA, 4 (1980), 1091–1107.
S. R. Bernfeld, P. Nagrini, and L. Salvadori, Quasi-invariant manifolds, stability and generalized Hopf bufurcation, Ann. Mat. Pure Appl. 150 (1982), 105–119.
N. Rouche, P. Habets and M. Laloy, Stability Theory by Liapunov’s direct method, Springer-Verlag, New York 1977.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1984 Springer-Verlag New York Inc.
About this chapter
Cite this chapter
Piccinini, L.C., Stampacchia, G., Vidossich, G. (1984). Questions of Stability. In: Ordinary Differential Equations in Rn . Applied Mathematical Sciences, vol 39. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5188-0_5
Download citation
DOI: https://doi.org/10.1007/978-1-4612-5188-0_5
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-90723-9
Online ISBN: 978-1-4612-5188-0
eBook Packages: Springer Book Archive