论文标题
通过频道设置值图生成有限不变的图：应用参数时间周期系统的稳定性分析
Generation of bounded invariants via stroboscopic set-valued maps: Application to the stability analysis of parametric time-periodic systems
论文作者
论文摘要
给出了一种方法，用于生成一个差异系统的有界不变的，该系统具有给定的一组初始条件$ x_0 $。这种不变的形式具有$ x_0 $始于欧拉的近似溶液上的管子，该元素的上限在大概解决方案和确切溶液之间的距离上。该方法包括查找真正的$ t> 0 $，以便在时间$ t =（i+1）t $的“快照”中包含在$ t = it $的快照中，对于某些整数$ i $。因此，在相空间中，不变的是圆环的形状。还提供了一个简单的附加条件，以确保系统的解决方案永远无法收敛到平衡点。在维度2中，这确保所有解决方案都趋向于极限周期。如果动态系统包含参数$ p $，则该方法将扩展，从而允许对系统的稳定性分析$ p $。这在古典范德尔的系统上进行了说明。
A method is given for generating a bounded invariant of a differential system with a given set of initial conditions around a point $x_0$. This invariant has the form of a tube centered on the Euler approximate solution starting at $x_0$, which has for radius an upper bound on the distance between the approximate solution and the exact ones. The method consists in finding a real $T>0$ such that the "snapshot" of the tube at time $t=(i+1)T$ is included in the snapshot at $t=iT$, for some integer $i$. In the phase space, the invariant is therefore in the shape of a torus. A simple additional condition is also given to ensure that the solutions of the system can never converge to a point of equilibrium. In dimension 2, this ensures that all solutions converge towards a limit cycle. The method is extended in case the dynamic system contains a parameter $p$, thus allowing the stability analysis of the system for a range of values of $p$. This is illustrated on classical Van der Pol's system.
