Dynamics in the Complex Plane
Explore what happens when simple mathematical functions are iterated repeatedly in complex numbers. Learn how orderly orbits give way to fractals, understand how critical points dictate stability, and navigate the Mandelbrot and Julia sets.
Like Duolingo, but for Dynamics in the Complex Plane. Tomo turns the whole topic into a game you play five minutes a day, until it actually sticks.
24 levels across 4 sections, about 48 minutes end to end, roughly 10 days at five minutes a day. It moves through Iterating in the Complex Plane, Classifying Local Stability and Critical Points, Navigating Parameter Space, and Rational Dynamics and Global Structure. It assumes you already know the basics.
Free forever · No credit card · iPhone & Android
Key ideas in Dynamics in the Complex Plane
- Iterating a function creates an orbit by feeding each output back as the next input
- Orbits can settle to a point, cycle periodically, wander chaotically, or escape
- Tracking orbit trajectories reveals whether a region remains bounded or diverges
- Fixed points occur where the function output equals the input without change
- Once an orbit hits a fixed point, every subsequent iteration stays on that point
- Fixed points act as the stationary structural anchors of the entire dynamical plane
- A derivative magnitude strictly less than one makes a fixed point attracting
- The complex derivative's argument introduces rotation at each successive step
- Combining contraction with rotation produces a spiral trajectory toward the point
- For z^2 + c with |c| <= 2, any orbit exceeding radius 2 escapes to infinity
- Beyond the escape radius, the squared term dominates all lower-order terms
- Once past the threshold radius, an orbit can never return to the bounded interior
- The escape-time algorithm counts iterations needed to exceed the escape boundary
- Different escape counts are mapped to a color palette to show rates of divergence
- Points that remain bounded up to the maximum iteration count form the interior
- The dynamical plane partitions completely into two complementary sets: the Fatou set and the Julia set.
You've tried the other tabs
Thirty open tabs. Four facts you actually kept.
You watched. You nodded. By Sunday it was gone.
One answer, then back to scrolling.
Eight weeks. You meant to finish. You didn't.
Tomo gives Dynamics in the Complex Plane the Duolingo treatment: levels, streaks, and quick quizzes that test what you just learned. That game loop is what the tabs above never had, so it's the one you actually finish.
Here's what playing it feels like
A real question from this course. Take your best guess.
Imagine following an orbit across the complex plane. What long-term paths can it take?
Get it right to open this lesson and 23 more in the app.
How to learn Dynamics in the Complex Plane, step by step
Go in order. 24 short levels across 4 sections, about 48 minutes in all, five minutes a day.
- 1
Iterating in the Complex Plane
- Tracing Orbits and Fixed Points
- Partitioning Order and Chaos: Fatou and Julia Sets
- 2
Classifying Local Stability and Critical Points
- Multipliers and Cycle Geometries
- The Central Role of Critical Points
- 3
Navigating Parameter Space
- The Mandelbrot Set as an Atlas
- Bifurcations and Bulbs
- 4
Rational Dynamics and Global Structure
- Newton's Method in the Complex Plane
- Global Topography on the Riemann Sphere
4 sections · 8 units · 24 levels. Built to play, not to enroll.
Questions about Dynamics in the Complex Plane
How do I learn Dynamics in the Complex Plane?
Go in order: start with Iterating in the Complex Plane, then Classifying Local Stability and Critical Points, then Navigating Parameter Space, then Rational Dynamics and Global Structure. On Tomo that's 24 short levels, about 48 minutes in all, played five minutes a day. Start free and play the first level in under a minute.
Is there a Duolingo-style app for Dynamics in the Complex Plane?
Yes. Tomo works like Duolingo: short daily lessons, streaks and quick quizzes. Dynamics in the Complex Plane is one of its free courses, and you can type any other topic to get a course of your own.
Is the Dynamics in the Complex Plane course free?
Yes. Dynamics in the Complex Plane is playable on Tomo's free plan, which doesn't expire. No credit card, no waitlist. Download the app and start your first lesson.
How long does it take to learn Dynamics in the Complex Plane?
24 levels across 4 sections, about 48 minutes of play in total. It's built for five minutes a day, so you finish it a little at a time instead of all at once.
Do I need any background to start Dynamics in the Complex Plane?
No. It's pitched at an intermediate level, so it starts where you are and builds up. If you can tap a phone, you can start.
Tomo is in ChatGPT
Tomo is also a ChatGPT plugin: the same playable courses, built and played inside your chat.
Add Tomo to ChatGPT- What to ask
- “Make me a Tomo course on black holes.” “Turn this chat into a Tomo course.”
- What you get
- Quick questions with instant feedback, hearts and stars, ready to play in about a minute.
- Where it goes
- No sign-up to start. Connect your Tomo account and your streak carries on in the app.
You pick the voice
Dynamics in the Complex Plane is taught in the Storyteller style: every lesson is a story. Want a different feel? In the app you can spin up the same topic in any of Tomo's teaching styles. Same facts, totally different vibe.
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Start Dynamics in the Complex Plane today.
Download Tomo, search Dynamics in the Complex Plane, and play your first lesson in under a minute.