#dynamical systems

[cauchyfeelings]

Solving stability proofs are actually really easy.

All you have to do is just guess and hope. If it’s a particularly difficult one I heard praying might help.

[noosphe-re]

SIAM Journal on Applied Dynamical Systems (SIADS) publishes research articles on the mathematical analysis and modeling of dynamical systems and its application to the physical, engineering, life, and social sciences. (https://epubs.siam.org/journal/sjaday)

[amhala]

Fixed-point

The basin of attraction in a dynamic manifold constitutes all possible initial conditions from which trajectories can start to explore the attractor in a coordinate phase space. These trajectories do not cross themselves usually and continue to trace dense orbits in the attractor inside the manifold. In certain manifolds, such as chaotic ones, they are characterized by sensitivity to their formation initial conditions so that they diverge rapidly from each other since that formation.

This process represented in essence a qualitative description of how dynamical systems evolve where classical descriptions, that included continuous mathematics such as calculus, provided the quantitative description most of the time in the form of analytical solutions. Then the need to conduct machine-based analysis brought about numerical techniques with varied degrees of accuracy but that relied on discrete, rather than continuous, mathematics.

Chaos theory was reborn again through such numerical analyses to provide not only qualitative treatment, but it came with an added set of mathematical tools that described natural phenomena and reproduced some of the important solutions found using classical mathematics. In addition, this numerical formulation of chaos theory allowed for toggling from continuous mathematics to discrete one, thereby giving an advantage when communicating in machine language to carry out computing jobs.

However, such communicating requires first the realization that the continuous form of chaotic attractors in their phase space must be discretized. So, these trajectories can by "discretized" into numerics before communicating their data to the computer. Many techniques exist to achieve this including the "collapse" of such continuous formulation into either a "fixed-point" or a "limit cycle". The first type is just one data point that does not change with mapping but reveals system stability, whereas the second formation is a confined regular attractor vs. the original so-called "strange attractor".

Strange attractor - it is a complex, fractal-shaped set of numerical values in a chaotic dynamical system that attracts nearby trajectories over time. Unlike simple attractors (points or loops), they exhibit extreme sensitivity to initial conditions leading to unpredictable, non-repeating behavior.

Nowadays, the dual utilization of qualitative and computational analyses of chaos theory gave it an advantage to understand many concepts over diverse disciplines that includes traditionally non-technological topics such as literary texts and societal sciences. In these contexts, in particularly, the terminology used in describing chaos theory concepts give direct access to further analyze them technically.

Figure: fixed points of functions in the complex plane commonly lead to patterned chaos fractal structures. The plots on the left color the value of the fixed point and on the right, they present the number of iterations to reach a fixed point for the Sine function.

[virtus-vincit-omnia]

[rauko]

3D Projection of Mandelbrot Set's Real Axis Bifurcation Diagram 𖣂

𝓏ₙ₊₁ = 𝓏ₙ² + 𝑐 -> 2D Mandelbrot Set, complex sequence notation

𝑓𝑐(𝑥) = 𝑥² + 𝑐 -> Bifurcation Quadratic Map, real function notation

[ladyfingerpress]

z_LXVIII "Pulse Makes Loops" or "Network Tensegrity: The Life Principle of Dynamic Systems" by Morgan Stern from Ladyfinger Press · https://archive.org/details/z_lxviii

[mndlbrt]

poisson saturne strange attractor

[a-lesson-you-never-asked-for]

Dynamical Systems

When we talk about dynamical systems, we broadly refer to the way a particular system evolves under certain conditions. For example, if you have a particle and you give it some terrain to roll around on, what path will it follow? If you have an electron and subject it to a magnetic field, what trajectory would it follow? Or if you impart forces on a block of water, how would the shape of the boundary change?

These are all types of questions that can be answered by dynamical systems, and involve a significant depth of analysis to truly understand their mechanics. (But truthfully, I've only ever been in it for the pictures).

The Tinkerbell map The Lorenz Attractor

As usual, if anyone has any feedback or errata to point out, please do shoot me a message :). I'm still getting back into the groove of things with this so I might be missing out on stuff.

Some basic definitions

We have 2 different types of trajectories, discrete and continuous.

discrete time dynamics - this means that you have observations at discrete points in time. You take a snapshot at t=1, t=2, etc. Take the Tinkerbell map above for example. You start with a point, then apply some sort of rule, and you end up at the next point. Things happen in steps.

continuous time dynamics - this means that at each point, you prescribe a velocity (a speed and direction) in which a particle should move. So if a particle is located at (0,0) for example, in an arbitrarily short period of time, it would move in a particular direction. A good example of this is the Lorenz map, as shown in the right hand image.

The Tinkerbell Map - equations of motion

The Lorenz Map - equations of motion

NOTE: In the first case, we talk about where the next point explicitly, whereas in the second case, we talk about in how fast in a particular direction we will end up moving.

A brief discussion on derivatives

Here's a visual to explain - suppose that the blue line represents how far away you are from some particular location. The average velocity would be the slope of the green line. But the instantaneous velocity would be equal to the the slope of the yellow line.

The way we denote this derivative is as follows

The numerator describes the change in the y-values, the denominator describes the change in the x-values, and the ratio of this is the gradient (or the slope).

So in a nutshell, the average velocity represents a change in distance over a period of time, and instantaneous velocity represents a change in distance of an arbitrarily short period of time.

Where to from here?

From here, we'll introduce some broad notation around maps, orbits, and some terminology that underpin dynamical systems. If there's enough time, I'll also describe some other concepts regarding different types of orbits, and thereafter we'll get into some really cool stuff.

Anyway, that's it for now! Adios :)