Change of Coordinates on an ODE
I have started this problem, but I'm confused as to what the problem wants me to do.
We are used to linearizing systems to extrapolate local behavior, which we can by taking the jacobian matrix at the equilibrium point.
I first noted that the jacobian is non-zero at the equilibrium point X_0, because it isn't the origin, that means that at least we can indeed create a linearization (change of coordinates).
But I'm not sure what to do from there...my guess is as follows.
Let B(X) be the transformation of coordinates (a matrix nxn), where B(X_0) = origin. Then we can approximate B(X) as B(X_0) + DF(X) + G(X), where DF is the jacobian at X_0 applied to X, and G(X) is an error term. G is suspiciously like the remainder term of a Taylor expansion, but I'm stuck here...
differential-equations
asked Nov 20 at 2:36
MathGuyForLife
1007
add a comment |
I have started this problem, but I'm confused as to what the problem wants me to do.
We are used to linearizing systems to extrapolate local behavior, which we can by taking the jacobian matrix at the equilibrium point.
I first noted that the jacobian is non-zero at the equilibrium point X_0, because it isn't the origin, that means that at least we can indeed create a linearization (change of coordinates).
But I'm not sure what to do from there...my guess is as follows.
Let B(X) be the transformation of coordinates (a matrix nxn), where B(X_0) = origin. Then we can approximate B(X) as B(X_0) + DF(X) + G(X), where DF is the jacobian at X_0 applied to X, and G(X) is an error term. G is suspiciously like the remainder term of a Taylor expansion, but I'm stuck here...
differential-equations
asked Nov 20 at 2:36
MathGuyForLife
1007
add a comment |
I have started this problem, but I'm confused as to what the problem wants me to do.
We are used to linearizing systems to extrapolate local behavior, which we can by taking the jacobian matrix at the equilibrium point.
I first noted that the jacobian is non-zero at the equilibrium point X_0, because it isn't the origin, that means that at least we can indeed create a linearization (change of coordinates).
But I'm not sure what to do from there...my guess is as follows.
Let B(X) be the transformation of coordinates (a matrix nxn), where B(X_0) = origin. Then we can approximate B(X) as B(X_0) + DF(X) + G(X), where DF is the jacobian at X_0 applied to X, and G(X) is an error term. G is suspiciously like the remainder term of a Taylor expansion, but I'm stuck here...
differential-equations
asked Nov 20 at 2:36
MathGuyForLife
1007
I have started this problem, but I'm confused as to what the problem wants me to do.
We are used to linearizing systems to extrapolate local behavior, which we can by taking the jacobian matrix at the equilibrium point.
I first noted that the jacobian is non-zero at the equilibrium point X_0, because it isn't the origin, that means that at least we can indeed create a linearization (change of coordinates).
But I'm not sure what to do from there...my guess is as follows.
Let B(X) be the transformation of coordinates (a matrix nxn), where B(X_0) = origin. Then we can approximate B(X) as B(X_0) + DF(X) + G(X), where DF is the jacobian at X_0 applied to X, and G(X) is an error term. G is suspiciously like the remainder term of a Taylor expansion, but I'm stuck here...
differential-equations
differential-equations
asked Nov 20 at 2:36
MathGuyForLife
1007
asked Nov 20 at 2:36
MathGuyForLife
1007
asked Nov 20 at 2:36
MathGuyForLife
1007
asked Nov 20 at 2:36
MathGuyForLife
1007
asked Nov 20 at 2:36
MathGuyForLife
1007
1007
add a comment |
add a comment |
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