A dynamical system is defined as follows:
$$\frac{dx}{dt}=\alpha x(1-y)(1+z)-x^2y$$
$$\frac{dy}{dt}=\beta y(1-z)(1+x)-y^2z$$
$$\frac{dz}{dt}=\gamma z(1-x)(1+y)-z^2x$$
For which set of (kind of) parameters $\alpha, \beta$ and $\gamma$, the system will possess chaos? How can we have an idea about parameters for local stability of the fixed points? How can we have Bifurcation Analysis of the parameters?
Let us compute the equilibria of the system by solving $\dot{X} = F(X) = 0$, where $X = (x,y,z)^\top$. The equilibrium equation rewrites as $$ \left\lbrace \begin{aligned} &x \left( \alpha(1-y+z-yz) -xy \right) = 0 \, , \\ &y \left( \beta(1-z+x-zx) -yz \right) = 0 \, , \\ &z \left( \gamma(1-x+y-xy) -zx \right) = 0 \, . \end{aligned} \right. $$ If $\alpha$, $\beta$ or $\gamma$ is zero, then there are equilibrium lines in $\mathbb{R}^3$, with expression $x=y=0$, $x=z=0$ or $y=z=0$. Thus, we assume that $\alpha$, $\beta$ and $\gamma$ are nonzero. Four equilibria are obtained analytically:
One last equilibrium corresponding to the case $x\neq 0$, $y\neq 0$, $z\neq 0$ must be considered, but it has no analytical expression. It could be obtained by solving numerically $$ \left\lbrace \begin{aligned} & 1-y+z-yz -xy/\alpha = 0 \, , \\ & 1-z+x-zx -yz/\beta = 0 \, , \\ & 1-x+y-xy -zx/\gamma = 0 \, . \end{aligned} \right. $$ Computing the eigenvalues of the Jacobian matrix $\partial F/\partial X$ at the equilibria, one can detect stable equilibria where all eigenvalues have a negative real part, Hopf bifurcations, Pitchfork bifurcations, etc. The transition to chaos may be more difficult to analyze, but can be observed numerically for some particular sets of parameters and initial conditions (see e.g. the case of the Lorenz system [1, 2]).