Differential Equation- Pendulum System

[Kamekui]

Homework Statement

(a). Find the critical points
(b). Linearize the system and determine local stability
(c). Find a Liapunov Function (Hint - Use total energy)
(d) Use the Liapunov Function to conclude that the critical points are stable.
(e.) Use the LaSalle theorem to argue that every trajectory has to tend toward one of the critical points.

Homework Equations

x'=y
y'=-y-sin(x)

The Attempt at a Solution

(a). Let x',y'=0
x'=0 → x=0

If x=0, then:

y'=0=0-sin(x)→ x=0

So, this implies there are infinite critical points occurring at (nπ,0) where n is an integer.


(b). Let x'=f(x,y), y'=g(x,y), the the Jacobian Matrix is
\begin{equation}

\left[
\begin{array}{ccc}
fx & fy \\
gx & gy
\end{array}
\right]= \left[
\begin{array}{ccc}
0 & 1 \\
-cos(x) & -1
\end{array}
\right]
\end{equation}

If n is even, then we get:

\begin{equation}

\left[
\begin{array}{ccc}
0 & 1 \\
-1 & -1
\end{array}
\right]

\end{equation}

\begin{equation}

\left[
\begin{array}{ccc}
x' \\
y'
\end{array}
\right]= \left[
\begin{array}{ccc}
0 & 1 \\
-1 & -1
\end{array}
\right]*\left[
\begin{array}{ccc}
x \\
y
\end{array}
\right]= \left[
\begin{array}{ccc}
y \\
-x-y
\end{array}
\right]

\end{equation}


The determinant yields: λ²+λ+1

→λ=-1/2-+ i(√3)/2

Since the eigenvalues are complex conjugates with negative real part, this implies the system is asymptotically stable locally at (nπ,0) (where n is even).

On the other hand, if n is odd we get: (I didn't do the Jacobian Matrix again because its a simple calc)

λ= -1/2+- (√5)/2

Since the eigenvalues are real and of opposite sign, this implies the critical points (nπ
,0) (where n is odd) are saddle points which means the critical point is locally unstable.


(c). This is where I get confused, finding Liapunov Functions is new to me. I think I'm supposed to assume V is a Liapunov Function of the form V = (1/2)ax² + (1/2)by².

So then ∇V= (2ax,2by)

v= ∇V ° (f,g)= axy-by²-by*sin(x)

So now I need to show that v≤kV, for some k<0

I'm lost, please provide any help you can, thank you

 

[mighty2000]

!

I will be happy to assist you in finding a Liapunov Function for this system. The Liapunov Function is a function that is used to analyze the stability of a system. It is defined as a scalar function of the system's state variables that decreases along the trajectories of the system.

In this case, we can use the total energy of the system as the Liapunov Function. The total energy of the system is given by E = (1/2)x'^2 + (1/2)y'^2 + y + cos(x).

We can show that this function decreases along the trajectories of the system by taking its derivative with respect to time:

dE/dt = x'x'' + y'y'' + y' + sin(x)x'

= x'(y' + sin(x)) + y'(y' + sin(x))

= (x')^2 + (y')^2 + y' + sin(x)x'

= (y')^2 + y' + sin(x)x' - (y')^2

= y' + sin(x)x'

= -(y + cos(x))y' + sin(x)x'

= -y'^2

Since y' is negative for all trajectories, this shows that dE/dt is always negative, and therefore the total energy E decreases along the trajectories of the system.

This implies that the critical points are stable, as the total energy decreases and tends towards 0 as time goes on.

Using the LaSalle theorem, we can argue that every trajectory of the system tends towards one of the critical points. This is because the total energy E decreases along the trajectories, and since it tends towards 0, the trajectories must tend towards the critical points where E = 0.

I hope this explanation helps you understand the concept of Liapunov Functions and their use in analyzing the stability of a system. If you have any further questions, please do not hesitate to ask. Good luck with your studies!