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An asymptotic solution slowly crossing the separatrix near a saddle-centre bifurcation point. (English) Zbl 1026.34062

The authors construct an asymptotic solution to the primary resonance equation \[ i\varepsilon U'+(|U|^2-t)U=1,\quad 0<\varepsilon\ll 1. \] The constructed solution has a special behaviour. This solution varies slowly when \(t>t_{*}\) and oscillates fast when \(t<t_{*}.\) The constant \(t_{*}\) defines the value of \(t\) when a saddle-centre bifurcation takes place. The asymptotic solution in a thin layer close to \(t_{*}\) is studied in detail by the matching method.

MSC:

34E13 Multiple scale methods for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
37G10 Bifurcations of singular points in dynamical systems
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