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.


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
Full Text: DOI


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