What is the long-term behavior of solutions for $dz/dt = 4z - z^3$ based on their initial conditions?

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

. Any initial condition strictly below

z = 0

will approach the stable equilibrium

z = -2

. The unstable equilibrium

z = 0

acts as a separatrix between these two basins of attraction.

Conditions

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

Reasoning, step by step

Identify the stable equilibria at $z = -2$ and $z = 2$ , and the unstable equilibrium at $z = 0$ .
Observe the direction of the vector field (arrows) on the phase line.
Note that for $z > 0$ , the flow is directed toward $z = 2$ .
Note that for $z < 0$ , the flow is directed toward $z = -2$ .
Conclude that $z(0) > 0$ implies $z(t) \to 2$ as $t \to \infty$ , and $z(0) < 0$ implies $z(t) \to -2$ as $t \to \infty$ .

Example

The speaker states, "initial conditions above zero would go to 2 and below zero would go to -2," summarizing the global behavior captured by the phase line diagram.

Common misconceptions

Assuming that all solutions converge to the same equilibrium.
Thinking that solutions starting exactly at $z = 0$ will move away over time (they stay at 0).
Confusing the direction of time with the direction of the spatial axis.

Watch the explanation

YouTubeBifurcations of a differential equation

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Connected concepts

Equilibria and stability →

What is the long-term behavior of solutions for $dz/dt = 4z - z^3$ based on their initial conditions?

Conditions

Reasoning, step by step

Example

Common misconceptions

Watch the explanation

Connected concepts

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Related questions

What is the long-term behavior of solutions for dz/dt=4z−z3dz/dt = 4z - z^3dz/dt=4z−z3 based on their initial conditions?

Conditions

Reasoning, step by step

Example

Common misconceptions

Watch the explanation

Connected concepts

Explore next

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What is the long-term behavior of solutions for $dz/dt = 4z - z^3$ based on their initial conditions?