On Existence of Infinitely Many Homoclinic Solutions
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Using the concept of an isolating segment, some sufficient conditions for the existence of homoclinic solutions to nonautonomous ODEs are obtained. As an application it is shown that for all sufficiently small \(\varepsilon >0\) there exist infinitely many geometrically distinct solutions homoclinic to the trivial solution \(z=0\) to the equation \[ \dot z=\bar z(1+\text{e}^{i\varepsilon t}|z|^2),\quad z\in\mathbb{C} . \]
- Jagiellonian University Poland
Averaging method for ordinary differential equations, Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics, homoclinic solutions, isolating segments, chaos, Homoclinic and heteroclinic solutions to ordinary differential equations, isolating segment, Periodic solutions to ordinary differential equations, fixed point index
Averaging method for ordinary differential equations, Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics, homoclinic solutions, isolating segments, chaos, Homoclinic and heteroclinic solutions to ordinary differential equations, isolating segment, Periodic solutions to ordinary differential equations, fixed point index
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