Mathematical Notes

, Volume 75, Issue 3–4, pp 352–359 | Cite as

Uniform Stability of Local Extrema of an Integral Curve of an ODE of Second Order

  • I. P. Pavlotsky
  • M. Strianese


A second-order equation can have singular sets of first and second type, S1 and S2 (see the introduction), where the integral curve x(y) does not exist in the ordinary sense but where it can be extended by using the first integral [1–-5]. Denote by Y the Cartesian axis y=0. If the function x(y) has a derivative at a point of local extremum of this function, then this point belongs to S1Y. The extrema at which y'(x) does not exist can be placed on S2. In [5–-8], the stability and instability of extrema on S1S2 under small perturbations of the equation were considered, and the stability of the mutual arrangement of the maxima and minima of x(y) on the singular set was studied (locally as a rule, i.e., in small neighborhoods of singular points). In the present paper, sufficient conditions for the preservation of type of a local extremum on the finite part of S1 or S2 are found for the case in which the perturbation on all of this part does not exceed some explicitly indicated quantity which is the same on the entire singular set.

ordinary differential equation integral curve local extremum singular point stability instability 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    I. P. Pavlotsky, M. Strianese, and R. Toscano, “Continuation of the solution of differential equations to a singular set,” Differentsialľnye Uravneniya [Differential Equations], 34 (1998), no. 3, 313–319.Google Scholar
  2. 2.
    I. P. Pavlotsky, M. Strianese, and R. Toscano, “Prolongation of the integral curve on the singular set via the first integral,” J. Interdisciplinary Math., 2 (1999), no. 2, 3, 101–119.Google Scholar
  3. 3.
    I. P. Pavlotsky and M. Strianese, “Extension of solution of ODE via the singular set,” Nonlinear Anal. (2002), no. 43, 4313–4317.Google Scholar
  4. 4.
    I. P. Pavlotsky and M. Strianese, Prolongation of Integral Curves of ODE of Second Order on the Singular Set via the Local First Integral, Preprint no. 1, Univ. of Salerno, Italy, 2002.Google Scholar
  5. 5.
    I. P. Pavlotsky, B. I. Sadovnikov, and M. Strianese, “Local stability of a singular set of the second type for a second-order ordinary equation,” Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.], 1 (2002), no. 3, 311–313.Google Scholar
  6. 6.
    I. P. Pavlotsky and M. Strianese, “On the stability of the singular set of a dynamical system,” Differentsialľnye Uravneniya [Differential Equations], 35 (1999), no. 3, 296–303.Google Scholar
  7. 7.
    I. P. Pavlotsky and M. Strianese, “Stability of a singular set of second-order ordinary differential equations with respect to εx-and εy-perturbations,” Dopov. Nats. Akad. Nauk Ukr., Mat. Prirodozn. Tekh. Nauki (2002), no. 4, 23–26.Google Scholar
  8. 8.
    I. P. Pavlotsky and M. Strianese, “On the local stability of the extrema of integral curves of second-order ordinary differential equations,” Mat. Zametki [Math. Notes], 71 (2002), no. 5, 742–750.Google Scholar
  9. 9.
    N. N. Bautin and E. A. Leontovich, Methods and Rules for the Qualitative Study of Dynamical Systems on the Plane [in Russian], Moscow, 1990.Google Scholar
  10. 10.
    Yu. I. Kaplun, V. Hr. Samoilenko, I. P. Pavlotsky, and M. Strianese, “The global theorem on implicit function and its applications in the theory of ordinary differential equations,” Dopov. Nats. Akad. Nauk Ukr., Mar. Prirodozn. Tekh. Nauki (2001), no. 6, 38–41.Google Scholar
  11. 11.
    G. Sansone and R. Conti, Equazioni differenziale non lineari, Cremonese, Roma, 1956.Google Scholar
  12. 12.
    I. P. Pavlotsky and G. Vilasi, “Some peculiar properties of relativistic oscillator in the post-Galilean approximation,” Phys. A, 214 (1995), 68–81.Google Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • I. P. Pavlotsky
  • M. Strianese

There are no affiliations available

Personalised recommendations