I have an exercise to linearize the following nonlinear system
$$\dot{x} = A - B x - x y^2$$ $$\dot{y} = A\cdot (x y^2 - y)$$
I tried it with
osz = NonlinearStateSpaceModel[{{A - B*x - x*y^2, A*(x*y^2 - y)}, {x, y}}, {x, y}, {A, B}]
for the input of this system.
How can I linearize this system now?
StateTransformationLinearize[osz]
doesnt obviously work, because its not an affine system...
a) To find equilibria, use Solve:
eq = Solve[{A - B*x - x*y^2, A*(x*y^2 - y)} == {0, 0}, {x, y}]
b) Linearizing around the point $(\hat x,\hat y)$ means making a new, linear system $$\dot{\vec{z}}=J\vec{z}$$ where $\vec{z}=(x-\hat x,y-\hat y)$ and $J$ is the Jacobian matrix $$J=\begin{bmatrix}d\dot x\over dx & d\dot x\over dy \\ d\dot y\over dx & d\dot y\over dy\end{bmatrix}$$ evaluated at $(\hat x,\hat y)$.
It's easy to calculate the Jacobian:
j = D[{A - B*x - x*y^2, A*(x*y^2 - y)}, {{x, y}}]
(* {{-B - y^2, -2 x y}, {A y^2, A (-1 + 2 x y)}} *)
The only thing left is to evaluate $J$ at the different equilibria:
j /. eq[[1]]
(* {{-B, 0}, {0, -A}} *)
j /. eq[[2]]
j /. eq[[3]]
To check the stability of the equilibria, calculate the eigenvalues of the Jacobian. If all eigenvalues have negative real part, then the equilibrium is stable. j/.eq[[1]] is easy to understand: the eigenvalues are on the diagonal. j/.eq[[2]] and j/.eq[[3]] are uglier. For those you might want to try the Routh-Hurwitz stability criteria: stable if $Tr(J)<0$ and $Det(J)>0$.
Sorry, I don't know about c).
References
Strogatz SH. 2014. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering.
Ellner SP, Guckenheimer J. 2006. Dynamic Models in Biology.
Your nonlinear system is a good example for showing Hopf and Bogdanov-Takens bifurcations. With the following changes
A=A0+y0;B=A0*y0;
we obtain the equilibria:
(*{{x -> 1/A0, y -> A0}, {x -> 1/y0, y -> y0}, {x -> (A0 + y0)/(A0 y0), y -> 0}}*)
If we analyze the linearization at the first non-trivial equilibrium
$$\left(\displaystyle\frac{1}{A_{0}},A_{0}\right)$$
we find that there is a Hopf bifurcation when $A_{0} = 1$ and $0<y_{0} <1$. Furthermore, if $A_{0} = 1$ and $y_{0}=1$, the two non-trivial equilibria collide at (1,1) and a Bogdanov-Takens bifurcation occurs.
The local stability analysis is easy with these changes.
NonlinearStateSpaceModelinStateSpaceModel(which is always linear) and it will convert automatically. Or you just define the non-linear equations inside ofStateSpaceModel. Look in the documentation ofStateSpaceModelunder "Scope" > "Basic Uses" for an example. $\endgroup$(x,y)point that you want to linearize around? Is the goal some kind of stability analysis? Of an equilibrium or of a limit cycle? $\endgroup$StateSpaceModel. There isnt a specific example which has something to do with my nonlinear system... @ChrisK: The exercise has three parts: a) find the stationary points b) linearize the system c) find a lyapunov-function I think you have to linearize this system with the stationary points, or at least with one of the two points. $\endgroup$