×

The stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator. (English) Zbl 1175.34053

The authors study the stability of a special type of nonlinear planar time-periodic system and generalise results of R. Ortega, called third order approximation, cf. [J. Dyn. Differ. Equations 4, No. 4, 651–665 (1992; Zbl 0761.34036), SIAM J. Math. Anal. 25, No. 5, 1393–1401 (1994; Zbl 0807.34065), J. Differ. Equations 128, No. 2, 491–518 (1996; Zbl 0855.34058), Ten mathematical essays on approximation in analysis and topology. Amsterdam: Elsevier. 215–234 (2005; Zbl 1090.34045)]. In particular, they consider the system
\[ \begin{cases} \dot{x}=a(t)y+c(t)y^{2n-1}+\frac{\partial G}{\partial y}(t,x,y),\\ \dot{y}=-b(t)x-d(t)x^{2n-1}-\frac{\partial G}{\partial x}(t,x,y), \end{cases}\tag{1} \] where \(a,b,c,d\) are \(T\)-periodic functions, and \(n\geq2\). Moreover, \(G: \mathbb R\times B_\epsilon(0)\rightarrow \mathbb R \) is a continuous, \(T\)-periodic function with continuous derivatives of all orders with respect to \((x,y)\), and \(G(t,x,y)=O((x^2+y^2)^{n+1/2})\) as \((x,y)\rightarrow 0\), uniformly with respect to \(t\in \mathbb R\).
The main result Theorem 3.5 is that if the trivial solution of the corresponding linear system problem is stable, \(\int_0^T |c(t)|\, dt\not=0\), \(\int_0^T |d(t)|\, dt\not=0\), and one of the following two conditions holds
(\(H_1\)) \(c(t)\geq 0\), \(d(t)\geq 0\),
(\(H_2\)) \(c(t)\leq 0\), \(d(t)\leq 0\),
then the trivial solution of (1) is stable.
The proof uses a reduction to a special case; they show that when a linear periodic planar Hamiltonian system is elliptic, there always exists a translation of time, which transforms the system into an \(R\)-elliptic one, i.e. the monodromy matrix of the transformed equation is a rigid rotation.
The main result is applied the relativistic oscillator.

MSC:

34C25 Periodic solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bereanu, C.; Mawhin, J., Existence and multiplicity results for some nonlinear problems with singular φ-Laplacian, J. differential equations, 243, 536-557, (2007) · Zbl 1148.34013
[2] Bereanu, C.; Mawhin, J., Periodic solutions of some nonlinear perturbations of the φ-Laplacian with possibly bounded φ, Nonlinear anal., 68, 1668-1681, (2008) · Zbl 1147.34032
[3] Chu, J.; Zhang, M., Rotation numbers and Lyapunov stability of elliptic periodic solutions, Discrete contin. dyn. syst., 21, 1071-1094, (2008) · Zbl 1161.37041
[4] Goldstein, H., Classical mechanics, (1980), Addison-Wesley Reading · Zbl 0491.70001
[5] Hale, J.K., Ordinary differential equations, (1980), Krieger New York · Zbl 0186.40901
[6] Lei, J.; Li, X.; Yan, P.; Zhang, M., Twist character of the least amplitude periodic solution of the forced pendulum, SIAM J. math. anal., 35, 844-867, (2003) · Zbl 1189.37064
[7] Levitan, B.M.; Sargsjan, L.S., Sturm – liouville and Dirac operators, Math. appl. (soviet ser.), vol. 59, (1991), Kluwer Academic Dordrecht
[8] Llibre, J.; Ortega, R., On the families of periodic orbits of the Sitnikov problem, SIAM J. appl. dyn. syst., 7, 561-576, (2008) · Zbl 1159.70010
[9] Liu, B., The stability of the equilibrium of reversible system, Trans. amer. math. soc., 351, 515-531, (1999) · Zbl 0924.58052
[10] Manásevich, R.; Mawhin, J., Periodic solutions for nonlinear systems with p-Laplacian-like operators, J. differential equations, 145, 367-393, (1998) · Zbl 0910.34051
[11] Manásevich, R.; Mawhin, J., Boundary value problems for nonlinear perturbations of vector p-Laplacian-like operators, J. Korean math. soc., 37, 665-685, (2000) · Zbl 0976.34013
[12] Moser, J., On invariant curves of area preserving mappings of an annulus, Nachr. akad. wiss. Göttingen math.-phys. kl. II, 1-20, (1962) · Zbl 0107.29301
[13] Núñez, D.; Ortega, R., Parabolic fixed points and stability criteria for nonlinear Hill’s equation, Z. angew. math. phys., 51, 890-911, (2000) · Zbl 0973.34046
[14] D. Núñez, P.J. Torres, KAM dynamics in the driven relativistic harmonic oscillator, preprint
[15] Ortega, R., The twist coefficient of periodic solutions of a time-dependent Newton’s equation, J. dynam. differential equations, 4, 651-665, (1992) · Zbl 0761.34036
[16] Ortega, R., The stability of equilibrium of a nonlinear Hill’s equation, SIAM J. math. anal., 25, 1393-1401, (1994) · Zbl 0807.34065
[17] Ortega, R., Periodic solution of a Newtonian equation: stability by the third approximation, J. differential equations, 128, 491-518, (1996) · Zbl 0855.34058
[18] Ortega, R., The stability of the equilibrium: A search for the right approximation, (), 215-234 · Zbl 1090.34045
[19] Ortega, R.; Zhang, M., Optimal bounds for bifurcation values of a superlinear periodic problem, Proc. roy. soc. Edinburgh sect. A, 135, 119-132, (2005) · Zbl 1088.34036
[20] Siegel, C.; Moser, J., Lectures on celestial mechanics, (1971), Springer-Verlag Berlin · Zbl 0312.70017
[21] Simo, C., Stability of degenerate fixed points of analytic area preserving mappings, Astérisque, 98-99, 184-194, (1982) · Zbl 0516.58028
[22] Sitnikov, K.A., Existence of oscillating motion for the three-body problem, Dokl. akad. nauk, 133, 303-306, (1960)
[23] Torres, P.J., Twist solutions of a Hill’s equations with singular term, Adv. nonlinear stud., 2, 279-287, (2002) · Zbl 1016.34044
[24] Torres, P.J., Existence and stability of periodic solutions for second order semilinear differential equations with a singular nonlinearity, Proc. roy. soc. Edinburgh sect. A, 137, 196-201, (2007) · Zbl 1190.34050
[25] Zhang, M., Sobolev inequalities and ellipticity of planar linear Hamiltonian systems, Adv. nonlinear stud., 8, 633-654, (2008) · Zbl 1165.34053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.