Problem understanding Center Manifold theory

Question

I am studying stability for non linear control systems, and I am focusing on the Center manifold theory .

In particular, I am trying to understand an example which is also in the Hassan K.Khalil book at pag. 311, and it is the following:

Consider the system:

$\dot{y} = yz$

$\dot{z}=-z+ay^{2}$

we have that the center manifold equation is:

$\dot{h}(y)[yh(y)]+h(y)-ay^{2}=0$

(1)

with boundary conditions :

$\dot{h}(y)=h(y)=0$

now, on the book it is said that this is hard to solve, so it is performed an approximation, and from this point I have doubts on how to proceed. I will say what I have understood so far so to explain better my doubts.

Since the equation of the center manifold is hard to solve, it will be done an approximation, by choosing:

$\dot{h}(y)=h_2(y)y^{2}+h_3(y)y^{3}+…$

and we will start firsr by considering $\dot{h}(y)\approx 0$ and if we cannot do considerations about the stability, we will use as approximation $\dot{h}(y)\approx h_2(y)y^{2}+O(|y|^3)$ and so on until we can say something about the stability at the origin.

In the example on the book, it is said that if I use $\dot{h}(y)\approx 0$, the reduced system is:

$\dot{y}=O(|y|^3)$

which, as far as I have understood, is obtaining sunstituting $\dot{h}(y)\approx 0$ into the center manifold equation (1), ans so the only non zero term that remains is $-ay^2$, so:

$\dot{y}=-ay^2+O(|y|^3)$

and it says that we cannot conclude nothing on the stability of the origin from here.

Why we cannot conclude anything?

Then, since we cannot conclude anything, it chooses $\dot{h}(y)\approx h_2(y)y^{2}+O(|y|^3)$,(2), and it says that if we substitute this into the center manifold equation (1), the reduced system is:

$\dot{y}=ay^3+O(|y|^4)$

the words used in the book to explain this are:

we substitute (2) into the center manifold equation and calculate $h_2$, by matching coefficients of $y^2$, to obtain $h_2=a$.

but, how did he get this result?

after this it says that for $a<0$ the origin is stable and for $a>0$ is unstable, but why?

I dont' understand some parts of this example, can somebody please help me?

Answer 1

1. You are misusing the dot derivative: it usually denotes the derivative with respect to time. For example, the dot derivative of a Lyapunov function $V(x)$ is a derivative with respect to time of the function $V(x(t))$, where $x(t)$ is a solution of the system: $\dot V(x)= \frac{dV(x(t))}{dt}$. (By the existence and uniqueness theorem, there is exactly one solution passing through any point $x$, therefore $\dot V(x)$ is correctly defined). This derivative is the rate of change of $V$ when moving along the trajectories of the system. If the function remains constant during movement, then $\dot V=0$. In the context of the center manifold theory, the derivative $h'(y)$ or $\frac{\partial h}{\partial y}$ is an ordinary derivative, not in the above sense. $\dot h(y)$ is a completely different thing: this is a derivative $\frac{dh(y(t))}{dt}$.
2. The approximation formula should be $$\tag{3} h(y)=h_2y^{2}+h_3y^{3}+\ldots $$ This is a Taylor expansion of an unknown function $h(y)$, so $h_2,h_3$ etc are constants.
3. There is no equation $\dot{y}=-ay^2+O(|y|^3)$ in the example. There are equations $\dot y=O(|y|^3)$ and $\dot y= ay^3+O(|y|^4)$ instead. Let's remember where the equation (1) came from. It was obtained from the change of variables $$ y=y,\quad w=z-h(y), $$ where $h(y)$ is an unknown function which satisfies the boundary conditions. In this variables our system can be written as $$\tag{4} \dot y= yz=y(w+h(y)) $$ $$\tag{5} \dot w= \dot z-h'(y)\dot y= -z+ay^2-h'(y)yz $$ $$ =-(w+h(y))+ay^2-h'(y)y(w+h(y)). $$ We want $w=0$ to be an invariant set of our system. For this we need that $w$ remains constant when traversing in the set $w=0$, i.e. $$ \dot w|_{w=0}=0. $$ Hence we get (from (5)) the condition $$ -(0+h(y))+ay^2-h'(y)y(0+h(y))=0 $$ or $$\tag{1} -h(y)+ay^2-h'(y)yh(y)=0. $$ This is the center mainfold equation for our system. If it is satisfied, then the motion of the original system on the manifold $w=0$ can be described by the reduced system $$\tag{4a} \dot y= yh(y). $$ According to Theorem 8.2, the stability type of the full system coincides with the stability type of the reduced system (4a). According to Theorem 8.3, we can substitute the first few terms of the Taylor expansion (3) into the reduced system to determine its stability. This is why in the example the expansion $h(y)=O(y^2)$ (not $h\approx 0$) is used first. The reduced system then takes the form $$\tag{4b} \dot y= y\cdot O(y^2)= O(y^3). $$ We cannot conclude anything from this equation because we have too little information about the system. This is why the author uses another expansion, $h(y)=h_2 y^2+O(y^3)$. In this case the reduced system is $$\tag{4c} \dot y= y(h_2 y^2+O(y^3))= h_2 y^3+ y O(y^3)=h_2 y^3+ O(y^4). $$ In order to find $h_2=a$, we should substitute $h(y)=h_2 y^2+O(y^3)$ into the center manifold equation (1). This part of the solution is described in the Cesareo's answer. Finally, we obtain the reduced system $$\tag{4d} \dot y= a y^3+O(y^4). $$
4. The reduced system (4d) is asymptotically stable if $a<0$ and unstable if $a>0$ because there is a Lyapunov function $V(y)= y^2$. Its derivative $\dot V= 2y\dot y= 2a y^4+O(y^5)$ is negative in some deleted neighborhood of the origin if $a<0$ and positive if $a>0$.