Figure \(\PageIndex{2}\): (a) The typical phase lines for \(y' = y^2 − \mu\). (b) Bifurcation diagram for \(y' = y^2 − \mu\). Figure \(\PageIndex{4}\): (a) Collection of phase lines for \(y^{\prime}=y...Figure \(\PageIndex{2}\): (a) The typical phase lines for \(y' = y^2 − \mu\). (b) Bifurcation diagram for \(y' = y^2 − \mu\). Figure \(\PageIndex{4}\): (a) Collection of phase lines for \(y^{\prime}=y^{2}-\mu y\). (b) Bifurcation diagram for \(y^{\prime}=y^{2}-\mu y\). The left one corresponds to \(\mu<\) 0 and the right phase line is for \(\mu>0\). (b) Bifurcation diagram for \(y^{\prime}=y^{3}-\mu y\).
Here \(\mu\) is a parameter that we can change and then observe the resulting effects on the behaviors of the solutions of the differential equation. When a small change in the parameter leads to larg...Here \(\mu\) is a parameter that we can change and then observe the resulting effects on the behaviors of the solutions of the differential equation. When a small change in the parameter leads to large changes in the behavior of the solution, then the system is said to undergo a bifurcation. The behavior of the solutions depends upon the sign of \(y^{2}-\mu y=y(y-\mu)\). For positive values of \(y\) we have that \(y' > 0\) and for negative values of \(y\) we have that \(y'<0\).
