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Assume that there is a vector field $\dot{x}=f(x),x\in \mathbb{R}^3$ with a Lyapunov-like function $V(x,y,z)$ and all the well-defined level set $\{V(x,y,z)=h\}$ is just a point or an ellipsoid(see the figure below). In addition, along the vector $\{\dot{V}=0\}$ also defines an ellipsoid $S$, and $\dot{V}$ satisfying that $$ \left\{ \begin{aligned} \dot{V}<0, ~~\text{if } x\in \text{exterior of } S \\ \dot{V}>0, ~~\text{if } x\in \text{interior of } S \\ \end{aligned} \right. $$ enter image description here

Then it seems that all the points transverse $S$ and then return to $S$ except the equilibrium points of vector field in $S$. Therefore, we can define a function $P:S\rightarrow S$, where $P(x)$ is defined as the first point returning to $S$ for the orbit starting at $x$. Additionaly, if $x$ is an equilibrium we define $P(x)=x$, and if $x\in S$ is also in the stable manifold of some equilibrium $e$ then we define $P(x)=e$.

So the function $P(x)$ is similar to the Poincare map, can we get some conclusions about the properties of the function $P(x)$, such as continuity?

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    Where is the point level-set located? Is it inside or outside of $S$? – 2017-02-20
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    @Evgeny sorry, the point level-set is just the bottom point of $S$. – 2017-02-21
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    By the way, I've tried sketching how vector field is arranged along the boundary of $S$. There are also options that vector field points strictly inwards or outwards along the boundary of $S$, and it is consistent with conditions of "Lyapunov-like" function. – 2017-02-22
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    It seems that the vector field transverses $S$ at all the points of $S$ except the equilibrium points. So $S$ is almost a poincare section. If we can get some properties about the flow-induced map on $S$, we can focus on $S$ to find some special orbits, such as periodic orbits. – 2017-02-25
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    Where exactly do you put equilibria on $S$? At the bottom point of $S$? Also why do you put them here? – 2017-02-25
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    oh, there may be several isolated equilibrium points on $S$. – 2017-02-26
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    @Evgeny Actually, I found this problem in the following equation: \begin{equation} \begin{aligned} &\dot{x}=-yz-0.06x-7.62y\\ &\dot{y}=-xz-0.02x-0.024y\\ &\dot{z}=xy-0.03z \end{aligned} \end{equation} The system above has three equilibrium points. Let $V(x,y,z)=\frac{x^2}{2}+\frac{y^2}{2}+7.64z+z^2$, then $S=\{(x,y,z):0.06x^2+0.024y^2+0.06z^2+0.03*7.64z=0\}$. – 2017-02-26
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    Oh, somehow I forgot that this system is not two-dimensional. By the way, is it important for you that your mapping is defined on the whole $S$? Frankly speaking, the continuity of this Poincare-like mapping can be done without requiring that it defined on the whole $S$. – 2017-02-26
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    yeah, it is not necessary to define the Poincare-like mapping on the whole $S$. But it seems that what I have known is just the recurrent behavior on the whole $S$. Since I don't know whether there exists a Poincare-like mapping on some true subset of $S$, I define the mapping on the whole $S$. Certainly, you can define the Poincare-like mapping on any suitable part of $S$. – 2017-02-27

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