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What is the equilibrium points of the system $$x'=yz-x^2 \quad , \quad y'=zx-y^2 \quad , \quad z'=xy-x^2$$ I also wondering how we can determine the trajectories of this system? can we somehow sketch them or imagine them?

Obviously the origin is one of the Equilibrium points. How we can find others? Is there any specific strategy for finding Equilibrium points?

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    @Moo Thanks for your help. what we can say about it's limit set. I mean like Omega-limit-set when $t \to \infty$. – 2017-01-03
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    Are you sure that the last equation is not $z'=xy-z^2$? – 2017-01-04
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    @JohnB I have checked that again it's $x^2$, But If you want to solve it with $z^2$ it's OK I just want to see how we treat with this kind of systems. – 2017-01-04

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How can we find other equilibrium points: Set $$x'=y'=z'=0$$ and then solve for $x,y,z$. This is the typical strategy. In this case start with $z'=x(y-z)=0$ i.e. $x=0$ or $y=z$, then plug this in the other two equations and so on.

Here you get the equilibrium points $(0,0,c)$ with $c \in \mathbb{R}$ and $(a,a,a)$ with $a \in \mathbb{R}$. So you have a whole range of equilibria; the whole $z$-axis consists of equilibria.

How to sketch the trajectories: You can either compute the solution (normally pretty hard) or you investigate the stability of your equilibria. You can look at Lyapunov functions to determine stability or look at the linearized system around an equilibrium $(x_0,y_0,z_0)$, i.e. $\xi'=J(x_0,y_0,z_0)\xi$ where

$$J(x,y,z)=\begin{pmatrix} -2x & z & y \\ z & -2y & x \\ y-2x &x &0 \end{pmatrix}.$$

Then plug in you equilibria and look for the eigenvalues. We call an equilibrium $(x_0,y_0,z_0)$ stable if $\text{Re}(\lambda)<0$ for all eigenvalues $\lambda$ of $J(x_0,y_0,z_0)$. Unstable if $\text{Re}(\lambda)>0$ for one eigenvalue. And if $\text{Re}(\lambda)=0$ for one eigenvalue then you can't say anything about stability with this method - one also says this is a non-hyperbolic equilibrium and this method (with the Grobman-Hartman-theorem) fails.

Now, for the eigenvalue $\lambda_i$ you can compute the according eigenvector $v_i$. If $\text{Re}\lambda_i<0$ then $v_i$ belongs to the stable eigenspace and if $\text{Re}\lambda_i>0$ then $v_i$ belongs to the unstable eigenspace.

Let me show you an example. You have the equilibrium $(0,0)$ and $\lambda_1=-1, \lambda_2=1, v_1=(1,0), v_2=(0,1)$ then your phase portrait looks like this:

enter image description here

You can see that the trajectories move according to the stable $\langle \binom{1}{0} \rangle$ and unstable $\langle \binom{0}{1} \rangle$ eigenspaces.

Since you liked this method I have drawn some other cases. They all have the same characteristic. First you look at the eigenspaces, draw them and then you draw your orbits according to them. Let's always take wlog $v_1=(1,0), v_2=(0,1)$ since otherwise just rotate your plane.

  • Upper left: $\lambda_1=\lambda_2>0$, $(0,0)$ is unstable
  • Upper right: $\lambda_1=\lambda_2<0$, $(0,0)$ is stable
  • Lower left: $0<\lambda_1<\lambda_2$, $(0,0)$ is unstable. Since $\lambda_1$ is nearer on the 'critical' zero and is in some sense 'less unstable' than $\lambda_2$, therefore the orbits move more towards $v_2$
  • Lower right: $0>\lambda_1>\lambda_2$, $(0,0)$ is stable

enter image description here enter image description here

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    The Jacobian is usually a good approach... except that, in the present case, at the equilibrium point $(0,0,0)$, every entry of the Jacobian matrix is zero hence one must find something else. // Your example at the end seems to be 2D hence the equilibrium should read $(0,0)$, not $(0,0,0)$. // Of course the typology of the fixed points of 3D systems such as the one the OP is interested in is much more complex. – 2017-01-07
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    @Did Thank you for you remark. Yeah the 3D case and the non-hyperbolic equilibrium $(0,0,0)$ makes this harder; I just wanted to show the general procedure/strategy. As I suggested one can also look for a Lyapunov function to determine stability (the OP asked in another post about Lyapunov functions so he knows how to compute them). Or one can look at neighborhood points i.e. take $(\epsilon, \epsilon, \epsilon)$ and look at the orbit if this point moves towards $(0,0,0)$ or not. – 2017-01-07
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    "the OP asked in another post about Lyapunov functions so he knows how to compute them" This is odd, effectively computing a Lyapunov function being notoriously hard. What are you alluding to? – 2017-01-07
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    @Did That it is possible to compute the stability of $(0,0,0)$ with a Lyapunov function. Not always easy, sure, but one could try with Polynomials $(ax+by+cz)^2$ and so on. I just mentioned it because the OP knows this concept and is familiar with it. – 2017-01-07
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    Maybe showing how to do that in the specific case at hand would be more convincing... – 2017-01-07
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    @Did I had the feeling that the OP does not know the general approach how to do this. So I gave him some ways so he can try. – 2017-01-07
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    Sorry but once again (and for the last time): *what do you call the general approach to compute a Lyapunov function?* – 2017-01-07
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    @Did I think we talk past each others. I meant: "I had the feeling that the OP does not know the general approach how to sketch the trajectories of a dynamical system. And I gave him ideas how to draw them." Not the general approach how to compute Lyapunov functions, there is none afaik, just clever trial and error, or one is lucky and has a Energy/Hamiltonian system. Then a Lyapunov function is just given by the Hamiltonian. – 2017-01-07
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    I just quoted what you wrote... But it is good that you made things clearer (and slightly different) in your last comment. – 2017-01-07
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    @MarvinF The idea of using eigenspaces was beautiful! I haven't seen it before in any book or something. I mainly used Cronin The Ordinary Differential Equation Theory which have few examples of how to find trajectories or main strategies to find Lyapunov functions. Could you pleas tell me any source which I can find more examples? – 2017-01-07
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    @shaha Happy to hear that :) I don't know your book but I have been working with Hirsch&Smale "Differential Equations, Dynamical Systems and an Introduction to Chaos", Wiggins "Introduction to Applied Nonlinear Dynamical Systems and Chaos" and Guckenheimer "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields". They are all great and have lots of examples, especially Hirsch/Smale uses a whole chapter how to draw phase portraits. – 2017-01-07
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    By the way, this system has continuous symmetry: if $(x(t), y(t), z(t))$ is a solution, then $(\lambda x(\lambda t), \lambda y(\lambda t), \lambda z(\lambda t))$ is also a solution for any $\lambda > 0$. – 2017-01-07
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    Here is why it could be useful: eventually any trajectory intersects (for example) unit sphere and if you are concerned with stability you have to consider how vector field behaves at this unit sphere. If all vectors point inside, then the equilibrium at the origin is asymptotically stable. One can use a level set of any homogeneous function to prove asymptotic stability (or its absence) this way. – 2017-01-07