Mixing
Can one use a Lyapunov-function to prove that an equilibrium is unstable?
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Is there a way to construct an argument that uses a Lyapunov-function to show that an equilibrium is unstable?
Let's consider the following example:
$$begin{align}dot x&=y+epsilon(x^3+2xy^2)\y&=-x+epsilon y^3end{align}$$
Then $V(x,y)=x^2+y^2$ is strictly positive on any neighbourhood of $(0,0)$ (with the exclusion of $(0,0)$ of course) and furthermore we have $$langlenabla V,f(x,y)rangle=2epsilon(x^2+y^2)^2$$ so it is also strictly decreasing along solutions if $epsilon<0$. Now, does this mean anything at all? Of course, for $epsilon>0$ this yields that $(0,0)$ is asymptotically stable. But can I use this function $V$ in the case I mentioned where $epsilon <0$?
ordinary-differential-equations stability-in-odes
asked Jan 26 at 11:37
RedLanternRedLantern
415
$endgroup$
add a comment |
$begingroup$
Is there a way to construct an argument that uses a Lyapunov-function to show that an equilibrium is unstable?
Let's consider the following example:
$$begin{align}dot x&=y+epsilon(x^3+2xy^2)\y&=-x+epsilon y^3end{align}$$
Then $V(x,y)=x^2+y^2$ is strictly positive on any neighbourhood of $(0,0)$ (with the exclusion of $(0,0)$ of course) and furthermore we have $$langlenabla V,f(x,y)rangle=2epsilon(x^2+y^2)^2$$ so it is also strictly decreasing along solutions if $epsilon<0$. Now, does this mean anything at all? Of course, for $epsilon>0$ this yields that $(0,0)$ is asymptotically stable. But can I use this function $V$ in the case I mentioned where $epsilon <0$?
ordinary-differential-equations stability-in-odes
asked Jan 26 at 11:37
RedLanternRedLantern
415
$endgroup$
$begingroup$
There are several instability theorems, the most powerful of which is the Chetaev theorem: en.wikipedia.org/wiki/Chetaev_instability_theorem. You can find the details and the proof in the book by H.Khalil, Nonlinear Systems (Theorem 4.3)
$endgroup$
– AVK
Jan 26 at 20:16
add a comment |
$begingroup$
Is there a way to construct an argument that uses a Lyapunov-function to show that an equilibrium is unstable?
Let's consider the following example:
$$begin{align}dot x&=y+epsilon(x^3+2xy^2)\y&=-x+epsilon y^3end{align}$$
Then $V(x,y)=x^2+y^2$ is strictly positive on any neighbourhood of $(0,0)$ (with the exclusion of $(0,0)$ of course) and furthermore we have $$langlenabla V,f(x,y)rangle=2epsilon(x^2+y^2)^2$$ so it is also strictly decreasing along solutions if $epsilon<0$. Now, does this mean anything at all? Of course, for $epsilon>0$ this yields that $(0,0)$ is asymptotically stable. But can I use this function $V$ in the case I mentioned where $epsilon <0$?
ordinary-differential-equations stability-in-odes
asked Jan 26 at 11:37
RedLanternRedLantern
415
$endgroup$
Is there a way to construct an argument that uses a Lyapunov-function to show that an equilibrium is unstable?
Let's consider the following example:
$$begin{align}dot x&=y+epsilon(x^3+2xy^2)\y&=-x+epsilon y^3end{align}$$
Then $V(x,y)=x^2+y^2$ is strictly positive on any neighbourhood of $(0,0)$ (with the exclusion of $(0,0)$ of course) and furthermore we have $$langlenabla V,f(x,y)rangle=2epsilon(x^2+y^2)^2$$ so it is also strictly decreasing along solutions if $epsilon<0$. Now, does this mean anything at all? Of course, for $epsilon>0$ this yields that $(0,0)$ is asymptotically stable. But can I use this function $V$ in the case I mentioned where $epsilon <0$?
ordinary-differential-equations stability-in-odes
ordinary-differential-equations stability-in-odes
asked Jan 26 at 11:37
RedLanternRedLantern
415
asked Jan 26 at 11:37
RedLanternRedLantern
415
asked Jan 26 at 11:37
RedLanternRedLantern
415
asked Jan 26 at 11:37
RedLanternRedLantern
415
asked Jan 26 at 11:37
RedLanternRedLantern
415
415
$begingroup$
There are several instability theorems, the most powerful of which is the Chetaev theorem: en.wikipedia.org/wiki/Chetaev_instability_theorem. You can find the details and the proof in the book by H.Khalil, Nonlinear Systems (Theorem 4.3)
$endgroup$
– AVK
Jan 26 at 20:16
add a comment |
$begingroup$
There are several instability theorems, the most powerful of which is the Chetaev theorem: en.wikipedia.org/wiki/Chetaev_instability_theorem. You can find the details and the proof in the book by H.Khalil, Nonlinear Systems (Theorem 4.3)
$endgroup$
– AVK
Jan 26 at 20:16
$begingroup$
There are several instability theorems, the most powerful of which is the Chetaev theorem: en.wikipedia.org/wiki/Chetaev_instability_theorem. You can find the details and the proof in the book by H.Khalil, Nonlinear Systems (Theorem 4.3)
$endgroup$
– AVK
Jan 26 at 20:16
$begingroup$
There are several instability theorems, the most powerful of which is the Chetaev theorem: en.wikipedia.org/wiki/Chetaev_instability_theorem. You can find the details and the proof in the book by H.Khalil, Nonlinear Systems (Theorem 4.3)
$endgroup$
– AVK
Jan 26 at 20:16
add a comment |
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$begingroup$
There are several instability theorems, the most powerful of which is the Chetaev theorem: en.wikipedia.org/wiki/Chetaev_instability_theorem. You can find the details and the proof in the book by H.Khalil, Nonlinear Systems (Theorem 4.3)
$endgroup$
– AVK
Jan 26 at 20:16