## 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

### Citations:

Zbl 0761.34036; Zbl 0807.34065; Zbl 0855.34058; Zbl 1090.34045
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### References:

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