On the fixed point index of the flow and applications to periodic solutions of differential equations on manifolds. (English) Zbl 0867.58055
The authors prove the formula
\[
\text{ind}(\Phi_T,\Omega)=\chi(-g,\Omega)\tag{1}
\]
where \(g: M\to \mathbb{R}^k\) is a tangent vector field on a boundaryless differentiable manifold \(M\subset \mathbb{R}^k\); \(\Omega\) is a relatively compact open subset of \(M\), and \(\chi(-g,\Omega)\) is the Euler characteristic (or index) of the vector field \(-g\) in \(\Omega\).
The authors apply formula (1) to the study of perturbed problems of \[ dx/dt=g(x)+\lambda f(t,x)\tag{2} \] where \(f:\mathbb{R} \times M\to \mathbb{R}^k\) is a \(T\)-periodic (with respect to the first variable) tangent vector field on \(M\). Furthermore, some results are obtained about the existence of \(T\)-periodic solutions of the perturbed equation (2).
The authors apply formula (1) to the study of perturbed problems of \[ dx/dt=g(x)+\lambda f(t,x)\tag{2} \] where \(f:\mathbb{R} \times M\to \mathbb{R}^k\) is a \(T\)-periodic (with respect to the first variable) tangent vector field on \(M\). Furthermore, some results are obtained about the existence of \(T\)-periodic solutions of the perturbed equation (2).
Reviewer: Di Zhang (Xian)
MSC:
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 37G99 | Local and nonlocal bifurcation theory for dynamical systems |
| 37C10 | Dynamics induced by flows and semiflows |
