How to use Python for scientific research

Mon, 14 Dec 2020 21:44:52 +0000

How to use Python for scientific research

Maintaining a personal Python package for scientific research

Our project should

Save data
Separate plotting from time-consuming code
Have good documentation
Be Distributable

Why Python?

It is personal preference however…

test

Auto-Animate

Layout

Root directory

exampleproject
│   README.md
│   setup.py
│
├───data
├───docs
├───exampleproject
├───notebooks
└───scripts

MA20221 Week 2

Mon, 14 Dec 2020 21:44:52 +0000

MA20221 Week 2

Definitions

What makes a well posed problem?

solution exists (for at least as long as the prediction is required)

- the solution is unique - the solution depends continuously on the initial condition and model parameters

What is a positively invariant set?

A set $Y \subset X$ is positively invariant if $x(0) \in Y$ implies that $x(t) \in Y$ for all $t \in T$ with $t > 0$. (It is negatively invariant if the same is true for all $t \in T$ with $t < 0$.)

Stability

Steady state: $x^*$, is where $f(x^*)=0$
Stable: for any $\epsilon > 0$ there exists $\delta > 0$ such that $|x(t) − x^*| < \epsilon$ for all positive $t \in T$ whenever $|x_0 − x^*| < \delta$, and unstable otherwise.
Asymptotically stable: if it is stable and there exists $\delta > 0$ such that $|x− x^*| \rightarrow 0$ as $t → \infty$ whenever $|x_0 − x^*| < \delta$.
stable: start close, stay close. asym. stable: start close, converge to steady state.

Time dependent solution

If $\frac{dx}{dt}=f(x)$ and $f(x_0)\neq0$, then $$ t = \int_{x_0}^{x(t)}\frac{ds}{f(s)} $$

If $x(t)$ is not a steady state, $x^*$,then it either tends to a steady state or it tends to $\pm\infty$
If the model system is well-posed, then $x(t)$ must either be a steady state solution, $x^*$, or strictly monotonic.

Slides | Jeremy Worsfold

How to use Python for scientific research