# Table of Contents for ME 2016 - Nonlinear Dynamical Systems 1

## Subdirectories
- [[2024-09-09/2024-09-09 - README]]
- [[images/images - README]]

## Files
- [[2024-08-26 Python Introduction.md]]
- [[2024-09-09 Example.py]]
- [[2024-09-09 Frameworks and Review.md]]
- [[2024-09-16 Plane Diagrams.md]]
- [[2024-09-18 Homework 1.md]]
- [[2024-09-23 More Phase Plane Stuff.md]]
- [[2024-09-30 Limit Cycles.md]]
- [[2024-10-28 Stability.md]]
- [[2024-11-11 Nonlinear 3D Phenomena.md]]
- [[2024-11-18 Volume Contraction.md]]
- [[2024-12-02 Delay Differential Equation.md]]
- [[ME2016_MEMS1016___Mini_Project_1.pdf]]
- [[Mini Project.md]]
- [[To Do - ME 2016.md]]

## Summary
It seems like you've provided a lot of information and notes on nonlinear dynamical systems, specifically focusing on stability and systems in the plane. I'll help summarize the main points and key concepts.

**Stability**

* Stability is about how systems behave around equilibrium points.
* **Lyapunov Stability**: A solution (equilibrium point) is stable if for any small disturbance, the system will return to its original state or converge to it.
* **Uniform Stability**: A solution is stable and the stability region does not change with time.
* **Asymptotic Stability**: A solution is asymptotically stable if it is stable and all trajectories converge to it as time approaches infinity.

**Systems in the Plane**

* Equilibrium points are fixed points where $\dot x = 0$.
* We can represent systems in the plane using diagrams, such as phase portraits or bifurcation diagrams.
* **Phase Portraits**: A diagram that shows the behavior of trajectories in the plane.
* **Bifurcation Diagrams**: A plot of equilibrium points vs. a parameter (e.g., amplitude).

**Plane Diagrams**

The following are some common types of plane diagrams:

1. **Limit Cycle**: A trajectory that oscillates around an equilibrium point.
2. **Linear Center**: An equilibrium point with orbits that move in a circular fashion around it.
3. **Periodic Point**: An equilibrium point where the system has periodic solutions.

These diagrams help us understand how systems behave near equilibrium points and can provide insights into their stability and dynamics.

**Poincaré Stability**

* Poincaré stability is a weaker concept than Lyapunov stability, which requires that all trajectories converge to an equilibrium point.
* A solution (equilibrium point) is Poincaré stable if it is stable and the distance between nearby trajectories remains bounded for small times.

**Unstable Systems**

* Unstable systems have trajectories that diverge from equilibrium points.
* Examples of unstable systems include:
	+ Simple harmonic oscillator
	+ Mass-spring system with a large spring constant

I hope this summary helps! Let me know if you'd like me to clarify any specific points or provide more information on these topics.

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