论文标题
无限尺寸中的退化Hopf分叉定理
A Degenerate Hopf Bifurcation Theorem in Infinite Dimensions
论文作者
论文摘要
为抽象进化方程$ \ frac {\ mathrm {d} x} {\ mathrm {d} t} = f（x，λ）$在无限尺寸下的hopf分叉定理是为$ re -rederm {\ mathrm {\ mathrm {\ mathrm {\ mathrm {d} = f（x，λ）$在归属感条件下的demeneracy demensions in denereracy dementions $ reμ^reμ^^^{\ prime} {\ prime}（\ piren^{\ pumper} {λ__________的= 0.0 $ and uppation。还得出了分叉周期溶液的稳定性。有趣的是，显示出跨危机分叉仍可以以$λ_0$发生，尽管小琐事解决方案的稳定性属性不会变化接近$λ_0$。我们的结果不需要$ f $的分析性。主要工具是Lyapunov- schmidt还原和摩尔斯式引理。多参数扩散捕食者的应用 - 纯化系统发现了周期性解决方案的新分支。
A Hopf bifurcation theorem is established for the abstract evolution equation $\frac{\mathrm{d}x}{\mathrm{d}t}=F(x,λ)$ in infinite dimensions under the degeneracy condition $Re μ^{\prime}(λ_0)= 0$ and suitable assumptions. The stability properties of bifurcating periodic solutions are also derived. Interestingly, it is shown that a transcritical Hopf bifurcation still can occur at $λ_0$ although the stability property of the trivial solutions does not change near $λ_0$. Our results do not require the analyticity of $F$. The main tools are the Lyapunov--Schmidt reduction and a Morse lemma. Applications to a multi-parameter diffusive predator--prey system discover new branches of periodic solutions.
