Averaging, Conley index continuation and recurrent dynamics in almost-periodic parabolic equations
Averaging, Conley index continuation and recurrent dynamics in almost-periodic parabolic equations
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DOI：
10.1016/j.jde.2004.07.008
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发表时间：
2004-02
影响因子：
2.4
通讯作者：
M. Prizzi
中科院分区：
文献类型：
--
作者：
M. Prizzi
研究了一类非自治抛物型方程，该方程具有几乎周期、快速振荡的主体和非线性相互作用。我们把这个方程与一个斜积半流联系起来，对于一类特殊的非线性，我们定义了孤立紧不变集的Conley指数。当振动的频率趋于无穷大时，我们证明了平均自治方程的每个孤立紧不变集可以继续到与非自治方程相关的斜积半流的孤立紧不变集。最后，我们给出了一些例子，在这些例子中，Conley指数可以被显式地计算出来，并且可以被用来检测方程中循环动力学的存在性。
We study a non-autonomous parabolic equation with almost-periodic, rapidly oscillating principal part and nonlinear interactions. We associate to the equation a skew-product semiflow and, for a special class of nonlinearities, we define the Conley index of isolated compact invariant sets. As the frequency of the oscillations tends to infinity, we prove that every isolated compact invariant set of the averaged autonomous equation can be continued to an isolated compact invariant set of the skew-product semiflow associated to the non-autonomous equation. Finally, we illustrate some examples in which the Conley index can be explicitly computed and can be exploited to detect the existence of recurrent dynamics in the equation.