I am desperate enough to ask this on f-ing tumblr, but nothing else has helped so far:
Does anyone know if the 2 equilibrium points (that are not (0,0,0)) of the Lorenz system are stable or not? And what is their classification (eg saddle point etc)?
I am going insane trying to figure it out but I just can't.
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.
An abuser denies the abuse ever took place, attacks the person that was abused (often the victim) for attempting to hold the abuser accountable for their actions, and claims that they are actually the victim in the situation, thus reversing what may be a reality of victim and offender. It often involves not just "playing the victim" but also victim blaming.
TL;Dr: Stop pathologizing neurodivergent people and individualizing abuse, and start treating abusers and bullies as a social failing that are products of privilege.
Unless you want to insist that every bitchass who's ever plagued marginalized people has NPD.
well what if the only person sqh doesn't know ANYTHING about is lqg because he was just there to be killed off and so he has to actually try to get to know him and can't use his author knowledge and so they bond and he sees lqg as so much more than just a character whose life was meant to end for a cheap shock and lqg also gets to know him as well and sees sqh in a new light because that coward he grew up distantly with is weirdly able to understand him and actually see him as a person and not just a dumb brute and a new friendship blossoms and
anyways i created the perfect ship/platonic ship name for them it's attack helicopter thank you for coming to my ted talk