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You Thought This Was Going To Be A Fanfiction But Actually It’s Just Perceptor Explaining Stability Theory To Shockwave

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Shockwave intimidatingly stood over Perceptor. “But you see, Autobot,” he wheezed, “I time travel to investigate chaos. That is a higher form of order and logic.”

“Oh, cool, I like chaos too,” said Perceptor. “That Lorentz attractor diagram is pretty rad.”

“Lorentz attractor?”

“Wait, you’re studying chaos, and you’ve never heard of a Lorentz attractor?”

Shockwave shook his gun intimidatingly, and looked around the room for his other gun, which looked like his alt-mode but smaller. Unable to locate it, he settled for just kind of shaking his arm. “Tell me of this artefact, autobot!” he demanded.

“Okay, so it’s this set of differential equations, x' = σ(y - x), and y' = x(ρ - z) - y, and z' = xy - βz when ρ is greater than one a pitchfork bifurcation occurs giving two new fixed points which for certain values of the parameters don’t correspond to periodic orbits, but instead to chaotic behaviour.”

Shockwave’s face lit up at the mention of chaos, like a dog near a doctor saying that a man’s son will never walk, before he realised that he didn’t understand any of the other words in the sentence. Looking around furtively to make sure that there were no other lifeforms to notice this embarrassing gap in his knowledge, he leaned towards Perceptor. “What’s a fixed point? And a bifurcation?” he said, quietly.

Perceptor cracked his robotic knuckles. “Okay, so are you familiar with differential equations?”

For a very long moment, Shockwave stared directly at Perceptor, optic to optic, entirely unmoving. Perceptor heard the quiet hum of regions of a Cybertronian processor, not used in millions of years, slowly crawling back into life, evil brain impulses flailing desperately to piece together the atrophied information. With correct validation techniques, even heavily degraded data could be restored, but looking at the strain that Shockwave was going through, Perceptor had his doubts.

“…yes?” he said, softly.

“Excellent! Okay, so imagine you have the differential equation, um, x' = -x. What happens over time when is initially positive?”

Shockwave thought for a moment. “It falls, to zero! Just as all of Cybertron shall!”

“…yeah, sure. Okay, but what if x is initially negative?”

“It rises to zero, just as all of Cybertron shall once I have finished placing ore around the galaxy!”

“And if it’s initially zero?”

“Well, it…” Shockwave paused. “Stays at zero?”

“Exactly! That’s what’s called a fixed point! And in the neighbourhood of the fixed point, the quantity moves towards it, so the fixed point is said to be stable.”
Shockwave grumbled. “I find stability very uninformative. What is there to gain from that which you can logically anticipate?”

Perceptor rubbed his hands together. “The next… let’s say thousand astroseconds… will be very aversive for you then.”

Shockwave checked his arm, which didn’t have any sort of timepiece on it. His motion, like many strange tics, was merely the skeuomorphic error of the great force animating all Cybertronian life. “I suppose I could tolerate it. After all, when I am done, I will have all the time on Cybertron. Continue with your talk of… stability.”

“Alright! If you have the time derivative of x be some general function x' = f(x), then you have a fixed point at f(x) = 0.”

“And these fixed points… they’re all stable?” Shockwave struggled to speak the word.

“Not quite!” said Perceptor. “Such a fixed point could be unstable.”

Shockwave rubbed his claws together in absolute glee. “How would you determine whether it was stable or not?”

“Ah, that’s simple, Shockwave. You simply look at the behaviour of the derivative when is varied a small amount from the fixed point.”

“And how would I do that?”

“How do you find the behaviour of any function near a certain point.”

Shockwave paused, then started pacing, then paused again, then did a strange repetitive motion as if the mercurial trickster forces were merely stalling for time. Smoke started wafting out of his cranial casing and Perceptor was briefly terrified that he had killed the bot. His optic flickered as power was diverted.

Finally, he tentatively spoke. “You… Taylor… expand?”

“Excellent work, Shockwave. Quite extraordinary! Yes, you Taylor expand. The value of the derivative at the fixed point is always zero, so the Taylor expansion is x' = f(x_0)(x - x_0). Now, solve it.”

“Solve… it?”

“First order differential equation! You can solve it!”

Shockwave raised his claws to his chin, making a remarkably accurate facsimile of the American Sign Language sign for lesbians, and held the pose for an uncomfortable number of frames. Then, he started clutching his head, in agony. He started shouting, and then stopped, and lowered his hands again. “The solution is an exponential.”

“You’re a quick one, Shockwave,” said Perceptor. He really wasn’t. “Specifically, (x - x_0) ~ exp(f'(x_0)t), which means that the difference grows exponentially if the derivative is positive and shrinks exponentially if the derivative is negative!”

“Corresponding to unstable and stable behaviour!”

“Yes, yes.”

“Good, but dull. I wish to make the stable unstable.”

“Then, Shockwave, you’ll be pleased to learn about bifurcations.”

“Bifur-whats?”

“Bifurcations! Imagine adding a parameter to your differential equation – like the k in x' = k - x^2.”

“Okay,” said Shockwave.

“What happens as this parameter varies?”

Shockwave paused, waiting for the answer.

“That was a question for you, Shockwave.”

“Oh, um…” said Shockwave.

“Find the fixed points.”

Shockwave thought. “Plus or minus the square root of k!”

“What’s their stability?”

Eternal moments as Shockwave attempted something as grand as a differentiation in his head. “Well, the derivative is -2x, which is negative at +sqrt(k), therefore stable, and positive at -sqrt(k), therefore unstable!”

“Yes, Shockwave! But is the square root of k always real?”

“If k is negative… there is no fixed point! The entire system is unstable!”

“Precisely. And at k = 0 we say there is a saddle node bifurcation – a bifurcation because we go from no fixed points, to two fixed points. How marvellous, Decepticon. Now, we can extend our arguments to two variables.”

“No!” yelled Shockwave, but he knew that the suffering had just begun.