Dynamical Systems
and Feedback

Computing and Data Science
A dynamical system is a system that
changes over time. Defined by:

  • State: variables representing current configuration
  • Rules: equations defining how the system evolves

Classic Dynamical Systems

  • Solar system
  • Weather systems
  • Economic systems
  • Chemical reactions
  • Electric Circuits
Dynamical systems are at the core of
science and the root of mathematics.

The primary question of a dynamical system is:
Can we predict future states?

Function

\[S(x) = \sqrt{x}\]

Function

\[S(x) = \sqrt{x}\]
\[ S(36) = 6 \] \[ S(20.25) = 4.5 \] \[ S(2) \approx 1.41421356237 \]

Function


Feedback


Feedback

\[S(x) = \sqrt{x}\]
\(x_0\)
\(x_1 = S(x_0) = \sqrt{x_0} \)
\(x_2 = S(x_1) = S(S(x_0)) = \sqrt{\sqrt{x_0}} \)
\(x_3 = S(x_2) = S(S(S(x_0))) = \sqrt{\sqrt{\sqrt{x_0}}} \)

Feedback

\[S(x) = \sqrt{x}\]
\(x_0\)
\(x_1 = S(x_0) = \sqrt{x_0} \)
\(x_2 = S(x_1) = S(S(x_0)) = \sqrt{\sqrt{x_0}} \)
\(x_3 = S(x_2) = S(S(S(x_0))) = \sqrt{\sqrt{\sqrt{x_0}}} \)
\(x_n = S^{n}(x_0) \)

Iterate

\[S(x) = \sqrt{x}\] \[\quad\] \(x_0 = 16\)
\(x_1 = S^1(x_0)=4\)
\(x_2 = S^2(x_0)=2\)
\(x_3 = S^3(x_0)=1.4142...\)
\(x_4 = S^4(x_0)=1.1892...\)
\(\vdots\)

Iterate

The orbit of \(x_0\) is the sequence generated by iterating
\[S(x) = \sqrt{x}\] \[\quad\] \(x_0 = 16\)
\(x_1 = S^1(x_0)=4\)
\(x_2 = S^2(x_0)=2\)
\(x_3 = S^3(x_0)=1.4142...\)
\(x_4 = S^4(x_0)=1.1892...\)
\(\vdots\)
This iterative function composition is equivalent to writing a recurrence relation with an initial condition: \[ x_0 = 16 \] \[ x_{n+1} = \sqrt{x_{n}} \]

Exercise

Determine the fate of the orbit of \(x_0=2\) in: \[F(x) = \sin(x)\]

In other words, what can we say about: \[\lim_{n \to \infty} F^{n}(2) \]

Determine the fate of the orbit of \(x_0=2\) in \[F(x) = \sin(x)\]

Exercise

Determine the fate of the orbit of \(x_0=2\) in: \[C(x) = \cos(x)\]

In other words, what can we say about: \[\lim_{n \to \infty} C^{n}(2) \]

Determine the fate of the orbit of \(x_0=2\) in \[C(x) = \cos(x)\]

Remember our algorithm for square root?


Algorithm — Square Root

Given a number \(x\), calculate \(\sqrt{x}\) to desired precision.


  1. Make a guess, \(g\).
  2. Calculate new guess: \(\frac{1}{2}( g + \frac{x}{g} )\)
  3. Repeate step (2) as many times as you like, plugging in the new guess each time.

Algorithm — Square Root

Given a number \(x\), calculate \(\sqrt{x}\) to desired precision.


  1. Make a guess, \(g\).
  2. Calculate new guess: \(\frac{1}{2}( g + \frac{x}{g} )\)
  3. Repeate step (2) as many times as you like, plugging in the new guess each time.

\[ x_0 = g \] \[ x_{i+1} = \frac{1}{2}\left(x_i + \frac{x}{x_i}\right) \]

Fixed Point

A fixed point is a value that stays the same from one step to the next in a dynamical system.
The output is the same as the input.

Given \(S(x)=\sqrt{x}\), \[S(1) = 1\]

Fixed Point

A fixed point is a value that stays the same from one step to the next in a dynamical system.
The output is the same as the input.

Given \(S(x)=\sqrt{x}\), \[S(1) = 1\]
\[S(0) = 0\]

Finding Fixed Points

\[\text{Output} = \text{Input}\] \[S(x) = x\]

Finding Fixed Points

\[\text{Output} = \text{Input}\] \[S(x) = x\]
\( \sqrt{x} = x\)
\( x = x^2 \)
\( x^2 - x = 0 \)
\( x(x-1) = 0 \)
\( x = 0, 1 \)
Recall our \(\sin\) and \(\cos\) systems.
Those convergence values were fixed points.
\[\sin(x) = x\]
 \(\quad \quad \)
\[\cos(x) = x\]

Periodic Orbit

A periodic orbit, or cycle, is a repeating sequence.

Finding Cycles


2-Cycle: \( \quad G(G(x)) = x \)

3-Cycle: \( \quad G(G(G(x))) = x \)

Finding Cycles


2-Cycle: \( \quad G(G(x)) = x \)

3-Cycle: \( \quad G(G(G(x))) = x \)

\(n\)-Cycle: \( \quad G^{n}(x) = x \)
Analysis of mathematical dynamical systems like these isn't just intellectually stimulating. We observe behaviors like this in real dynamical systems.

  • Physical systems: orbits(!), phases, tides
  • Weather: seasons, atmospheric convection, water cycles
  • Population: illness, predator-prey, consequence of policies, external factors like changes to environment
  • Psychology: modes of cognition and learning, response to stimuli
  • Politics: pendulum swing of political parties, cultural waves
  • Electronics: harnessing feedback, limit cycle, amplification

Pretty much everything is a dynamical system with feedback...Ok not everything

Population Dynamics

\[ P_{n+1} = R P_{n} \]

Population Dynamics

\[ P_{n+1} = R P_{n} \]
\(R\) is a parameter representing growth-rate.
We analyze this system by looking at its behavior for various values of \(R\text{.}\)

What are interesting values of \(R\text{?}\)
\( A_{n+1} = 1.1 A_{n} \)
\( B_{n+1} = 1.0 B_{n} \)
\( C_{n+1} = 0.9 C_{n} \)

Slightly Less Trivial Population Dynamics

\[P_{n+1} = r P_{n} (1 - P_{n}) \]

Slightly Less Trivial Population Dynamics

\[P_{n+1} = r P_{n} (1 - P_{n}) \]

\(r\) is a parameter that encapsulates growth and carrying capacity. \(P\) is a population density.

Slightly Less Trivial Population Dynamics

\[P_{n+1} = r P_{n} (1 - P_{n}) \]

Analyze this system for:
\( r = 2.5 \)
\( r = 3 \)
\( r = 3.5 \)
\( r = 4 \)
\(P_{n+1} = r P_{n} (1 - P_{n}) \)
\(\quad \)

Determinism

Given a dynamical system and inital conditions, if I ask you to find the \(83^{\text{rd}}\) term, you should all get exactly the same answer.
That is deterministic. That is not random.

Chaos

A system is chaotic if it exhibits
sensitive dependence on initial conditions.
\(x_0 = 8 \)
\(x_0 = 8.00001\)












https://twitter.com/standupmaths/status/741251532167974912
Chaos has some of the best lore!

Deterministic Vs. Predictable

But, even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon.
Poincaré

Complexity Science

In chaos, we see simple systems exhibit infinitely complex behaviors.

Complexity science is in some ways a converse of chaos: Profoundly complex systems may exhibit order.
https://www.complexityexplorer.org/explore/glossary

Dynamical Systems

Feedback The output of a system is fed back into the input
Iteration One step of a feedback loop
Orbit The sequence generated by iterating from a particular initial condition
Fixed Point A value such that the output of an iteration is equal to the input
Cycle A periodic orbit, repeating values
Deterministic When a system does not involve randomness, and future states can be calculated exactly.
Chaotic When the orbit of a dynamical system is sensitively dependent on the initial conditions.
A fascinating history of Chaos! Check out Gleick's book, and
Strogatz's historical overview and Ross's
1666 Newton Dynamics of motion. Differential calculus.
~ 1900 Poincaré Geometric view of dynamics. Shows 3-body problem is intractable.
1920s - 50s Nonlinear oscillators in physics and engineering.
~1950 Modern computing develops.
1963 Lorenz "Deterministic nonperiodic flow"
1976 May "Simple mathematical models with very complicated dynamics"
1970s Mandelbrot Fractals
1978 Feigenbaum Universal route to chaos
1999 Williams "Nonlinear Grid Ciphers", my science fair project on chaos for cryptography!

References and Resources

©2025 Jedediyah Williams
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