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The long-term behavior is determined by the phase line. Any initial condition strictly above the unstable equilibrium $z = 0$ will approach the stable equilibrium $z = 2$.

Conditions: The differential equation is $dz/dt = 4z - z^3$.; Initial conditions are not exactly at the unstable equilibrium $z = 0$.

The long-term behavior is determined by the phase line. Any initial condition strictly above the unstable equilibrium $z = 0$ will approach the stable equilibrium $z = 2$.

Conditions: The differential equation is $dz/dt = 4z - z^3$.; Initial conditions are not exactly at the unstable equilibrium $z = 0$.

A bifurcation diagram is a plot that shows how the equilibrium values of a dynamical system change as a parameter varies. It is constructed by plotting the equilibrium values (z) on the vertical axis against the parameter (C) on the horizontal axis.

Conditions: The system is described by a family of ODEs parameterized by C.; Equilibria are determined for a range of C values.

A bifurcation diagram is a plot that shows how the equilibrium values of a dynamical system change as a parameter varies. It is constructed by plotting the equilibrium values (z) on the vertical axis against the parameter (C) on the horizontal axis.

Conditions: The system is described by a family of ODEs parameterized by C.; Equilibria are determined for a range of C values.

Equilibria correspond to the roots of the right-hand side function $f(z) = 4z - z^3 + C$. As C increases, the graph of $f(z)$ shifts vertically upward.

Conditions: The differential equation is $dz/dt = 4z - z^3 + C$.; $C = 4$, which is greater than the critical value C* ≈ 3.079.

Equilibria correspond to the roots of the right-hand side function $f(z) = 4z - z^3 + C$. As C increases, the graph of $f(z)$ shifts vertically upward.

Conditions: The differential equation is $dz/dt = 4z - z^3 + C$.; $C = 4$, which is greater than the critical value C* ≈ 3.079.

For $C = 4$, the system has only one stable equilibrium. Because there are no other equilibria to act as separatrices, all initial conditions, regardless of their starting value, will converge to this single positive equilibrium as time progresses.

Conditions: The differential equation is $dz/dt = 4z - z^3 + 4$.; $C = 4$ is greater than the bifurcation threshold C* ≈ 3.079.

For $C = 4$, the system has only one stable equilibrium. Because there are no other equilibria to act as separatrices, all initial conditions, regardless of their starting value, will converge to this single positive equilibrium as time progresses.

Conditions: The differential equation is $dz/dt = 4z - z^3 + 4$.; $C = 4$ is greater than the bifurcation threshold C* ≈ 3.079.

Stability is determined by the direction of the vector field, which is given by the sign of $dz/dt$. Near $z = -2$ and $z = 2$, the arrows on the phase line point toward these equilibria, meaning nearby trajectories converge to them, making them stable.

Conditions: The system is a one-dimensional autonomous ODE.; Stability is inferred from the sign of $dz/dt$ on intervals adjacent to the equilibria.

Stability is determined by the direction of the vector field, which is given by the sign of $dz/dt$. Near $z = -2$ and $z = 2$, the arrows on the phase line point toward these equilibria, meaning nearby trajectories converge to them, making them stable.

Conditions: The system is a one-dimensional autonomous ODE.; Stability is inferred from the sign of $dz/dt$ on intervals adjacent to the equilibria.

The critical parameter value is approximately C* = 3.079. Below this value ($C < C$*), the differential equation has three equilibria.

Conditions: The differential equation is $dz/dt = 4z - z^3 + C$.; C* is the threshold where the number of real roots of $4z - z^3 + C = 0$ changes.

The critical parameter value is approximately C* = 3.079. Below this value ($C < C$*), the differential equation has three equilibria.

Conditions: The differential equation is $dz/dt = 4z - z^3 + C$.; C* is the threshold where the number of real roots of $4z - z^3 + C = 0$ changes.