论文标题
鞍节分叉的正常形式：塔琴的系数和在气候模型中的应用
Normal forms for saddle-node bifurcations: Takens' coefficient and applications in climate models
论文作者
论文摘要
We show that a one-dimensional differential equation depending on a parameter $μ$ with a saddle-node bifurcation at $μ=0$ can be modelled by an extended normal form $\dot y = ν(μ)-y^2+a(μ)y^3$, where the functions $ν$ and $a$ are solutions to equations that can be written down explicitly.对原始方程的等效性是吸引人盆地上的局部可区分共轭性，并在其存在的参数区域中固定点排斥，否则在整个局部间隔上都是可区分的共轭。 （回想一下，在标准方法中，局部等效性是拓扑而不是可区分的。）$ a（0）$是正常形式理论中的takens系数。 结果解释了正常形式从分叉点延伸的意义，并提供了鞍形节点分叉的新的，更详细的表征。一维系统可以使用中心歧管理论从较高的维方程得出。我们使用气候科学的两个示例来说明这一点，并展示了如何在某些设置中分析确定功能$ν$和$ a $，而在其他设置中可以在数值上确定。
We show that a one-dimensional differential equation depending on a parameter $μ$ with a saddle-node bifurcation at $μ=0$ can be modelled by an extended normal form $\dot y = ν(μ)-y^2+a(μ)y^3$, where the functions $ν$ and $a$ are solutions to equations that can be written down explicitly. The equivalence to the original equations is a local differentiable conjugacy on the basins of attraction and repulsion of stationary points in the parameter region for which these exist, and is a differentiable conjugacy on the whole local interval otherwise. (Recall that in standard approaches local equivalence is topological rather than differentiable.) The value $a(0)$ is Takens' coefficient from normal form theory. The results explain the sense in which normal forms extend away from the bifurcation point and provide a new and more detailed characterisation of the saddle-node bifurcation. The one-dimensional system can be derived from higher dimensional equations using centre manifold theory. We illustrate this using two examples from climate science and show how the functions $ν$ and $a$ can be determined analytically in some settings and numerically in others.
