The Stability and Instability of Steady States

TL;DR

The video discusses the concept of stability and instability of steady states in differential equations through examples and visual demonstrations.

Transcript

GILBERT STRANG: This is a topic I think is interesting. I like this one. It's about stability or instability of a steady state. So let me show you the differential equation. It could be linear, but might be non linear. dy dt is f of y. I'm going to-- I keep it that right hand side not depending on t, so just a function of y. And when do I have a st... Read More

Key Insights

• 🫱 The stability or instability of a steady state in a differential equation is determined by the sign of the derivative of the right-hand side of the equation at that steady state.
• ❎ In linear equations, the stability of the steady state depends on whether the constant in the equation is negative or positive.
• 🫱 The logistic equation has two steady states, and their stability depends on the value of the right-hand side of the equation at those steady states.
• 🫢 The equation y - y^3 = 0 has three steady states, and their stability can be determined by the sign of the derivative of the right-hand side at those steady states.
• 📔 The concept of stability and instability can be easily understood by visual demonstrations such as throwing a book and observing its motion in different directions.
• 🏑 The stability or instability of steady states has practical applications in various fields such as physics and biology.
• 🖐️ Calculus plays a crucial role in understanding the difference between steady states and their behavior as the solution approaches them.

Questions & Answers

Q: What is the criteria for determining whether a steady state is stable or unstable?

To determine stability, we look at the derivative of the right-hand side of the equation at the steady state. If the derivative is negative, the steady state is stable. If it is positive, the steady state is unstable.

Q: How does stability or instability affect the behavior of the solution near the steady state?

If a steady state is stable, the solution starting near it will approach the steady state as time goes on. On the other hand, if a steady state is unstable, the solution starting near it will move away from the steady state.

Q: What are the steady states and their stability for the logistic equation?

The logistic equation has two steady states: 0 and 1. The steady state at 0 is unstable, while the one at 1 is stable. Solutions starting at 0 move away from it, while solutions starting at 1 approach it.

Q: How many steady states are there in the equation y - y^3 = 0?

The equation y - y^3 = 0 has three steady states: -1, 0, and 1. The steady state at 0 is unstable, while the ones at -1 and 1 are stable. Solutions starting near -1 or 1 approach their respective steady states.

Summary & Key Takeaways

• Stability and instability of a steady state in a differential equation depend on whether the derivative of the right-hand side of the equation is negative or positive at the steady state.

• In linear equations, a steady state is stable if the exponential of the constant in the equation is negative.

• In the logistic equation, there are two steady states: 0 and 1. The steady state at 0 is unstable, while the one at 1 is stable.

• In the equation y - y^3 = 0, there are three steady states: -1, 0, and 1. The steady state at 0 is unstable, while the ones at -1 and 1 are stable.