## Differential and Integral Equations

- Differential Integral Equations
- Volume 9, Number 5 (1996), 1165-1181.

### Exponentially small bifurcation functions in singular systems of O.D.E

#### Abstract

We consider the singularly perturbed problem (*) $$\begin{cases} \epsilon\dot u & = Ju+\epsilon^p g(u,v,\epsilon), \\ \dot v & = f(u,v,\epsilon) \end{cases} $, $J= \begin{pmatrix} 0 &-1 \\ 1 &0 \end{pmatrix}$$, assuming that the degenerate system $\dot v=f(0,v,0)$ has an orbit $v_0(t)$ homoclinic to the hyperbolic equilibrium $v=0$. Under certain conditions on $f, g$ (that are stated in Section 2), we show that the bifurcation towards a homoclinic orbit $v(t,\epsilon)$ of (*) depends on three bifurcation functions $G_i(\alpha ,\epsilon), \, i=1,2,3, \, \alpha \in \mathbb{C} $, that are $2\pi \epsilon$-periodic in $\alpha $, for $\epsilon\neq 0$, and satisfy $|G_i(\alpha ,\epsilon)-G_i^0(\epsilon)| \leq C\epsilon^{-1} e^{-\eta _0/|\epsilon|}$, $\eta _0 >0$, where $G_i^0(\epsilon)={1\over {2\pi \epsilon}}\int_{0}^{2\pi \epsilon} G_i(\alpha ,\epsilon)\, d\alpha$. Thus we see that if $G_i^0(\epsilon)=0$ the bifurcation functions are exponentially small. This fact is then used to recover some of the results shown in [4].

#### Article information

**Source**

Differential Integral Equations, Volume 9, Number 5 (1996), 1165-1181.

**Dates**

First available in Project Euclid: 6 May 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1367871537

**Mathematical Reviews number (MathSciNet)**

MR1392101

**Zentralblatt MATH identifier**

0853.34035

**Subjects**

Primary: 34C37: Homoclinic and heteroclinic solutions

Secondary: 34C23: Bifurcation [See also 37Gxx] 34E15: Singular perturbations, general theory

#### Citation

Battelli, Flaviano. Exponentially small bifurcation functions in singular systems of O.D.E. Differential Integral Equations 9 (1996), no. 5, 1165--1181. https://projecteuclid.org/euclid.die/1367871537