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We study the behavior of a continuous flow near a boundary. We prove that if φ is a flow on $E={ℝ}^{n+1}$ for which $\partial E={ℝ}^n\times 0$ is an invariant set and S ⊂ ∂E is an isolated invariant set, with non-zero homological Conley index, then there exists an x in EE such that either α(x) or ω(x) is in S. We also prove an index theorem for a flow on ${ℝ}^n\times [0,\infty )$.

We show that under some assumptions on the function f the system $ż=z̅(f\left(z\right)e^{i\varphi t}+e^{i2\varphi t})$ generates chaotic dynamics for sufficiently small parameter ϕ. We use the topological method based on the Lefschetz fixed point theorem and the Ważewski retract theorem.

We prove that the Poincaré map ${\phi }_{(0,T)}$ has at least $N(\tilde{h},cl\left(W_0{W}_0^{-}\right))$ fixed points (whose trajectories are contained inside the segment W) where the homeomorphism $\tilde{h}$ is given by the segment W.