In two dimensions, if we have a dynamical system: $$\dot{x}=f_k(x,y)$$ $$\dot{y}=g_k(x,y)$$ with $f$ and $g$ smooth functions and $k$ is a paremeter. If $k=k^*$ is a bifurcation at which two equilibrium points, one stable and another unstable, coalesce. Is it true that for that particular $k$ there is an homoclinic orbit?
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No.
Consider the two-dimensional dynamical system, dependent upon the parameter $k$, given by
$\dot x = x(x - k), \tag{1}$
$\dot y = -y, \tag{2}$
and take $k > 0$. This system has two equilibria, one at $(0, 0)$ and one at $(k, 0)$. The equilibrium point at $(0, 0)$ is stable and the one at $(k, 0)$, being a saddle, is unstable. For every $k > 0$, there is a heteroclinic orbit $\gamma_k(t) = (x_k(t), 0)$ such that $\lim_{t \to -\infty} \gamma(t) = (k, 0)$ and $\lim_{t \to \infty} \gamma(t) = (0, 0)$. It is easy to see that as $k \to 0^+$, the position of the saddle shifts to the left; when $k = k^* = 0$, the heteroclinic orbit momentarily disappears as the equilibria coalesce; but in no case is there a homoclinic trajectory joining $(0, 0)$ to itself, unless the trivial orbit $\gamma_0(t) = (0, 0)$ is admitted as homoclinic; but this is, to my knowledge, against all customary usage of the term.
Hope this helps. Cheerio,
and as always,
Fiat Lux!!!
Robert Lewis
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