Chaos and Nonlinearity

Chaos

The word chaos comes from Latin chaos, derived from Greek cháos, meaning an infinite empty space or formless primal matter.

• a state of complete disorder or confusion
• lacking apparent structure or organization

→ Many cosmogonic myths begin with chaos.

Chaos theory

Chaos theory studies nonlinear dynamical systems and explores how simple deterministic rules can lead to complex, unpredictable behavior.

Examples:

• Weather and climate models (Lorenz attractor)
• Population dynamics (logistic map)
• Celestial mechanics (three-body problem)

Dynamical systems

Systems that evolve and change over time according to defined rules.

• Defined by equations or algorithms (e.g., differential equations)
• Can be linear or nonlinear
• Behavior can be predictable or chaotic

Nonlinearity

The output does not change proportionally with the input.

• All chaotic systems are nonlinear
• But not all nonlinear systems are chaotic

→ Nonlinearity is a fundamental prerequisite for chaotic behavior.

Sensitivity to initial conditions

Minimal differences in initial conditions and small changes can lead to unpredictable shifts in system states.

→ Butterfly effect (Edward N. Lorenz (1917 - 2008))

Deterministic chaos

The system follows deterministic rules, but small uncertainties in its initial conditions limit reliable long-term prediction.

Example: Chaos game

Iteration rule:

, and are corners of an equilateral triangle
random number 1, 2, or 3
random starting point

→ Draw the points to

Emergence of fractal patterns

An iteration rule defines how a system transitions from one state to the next:

• A simple shape evolves into a complex form through recursion.

→ This process leads to self-similarity

Emergence

• System-level behavior not obvious from parts

• Examples: flocking birds

Chaos and emergence

Chaos and emergence are related but distinct:

• chaotic dynamics can generate complex patterns
• emergence describes system-level behavior arising from interactions

Population dynamics and the logistic map

The logistic map is a classic example of how iteration rules can produce complex and chaotic behavior as well as fractal structures.

It uses the formula:

• : current state (e.g., population),
• : growth rate parameter

→ Behavior ranges from stability to chaos as increases

Logistic map behaviors

For , the logistic map exhibits several characteristic behaviors:

a) Extinction: dies away to zero
b) Stable fixed point: settles on one value
c) Period-doubling cascade: cycles of increasing period
d) Chaos with periodic windows

→ These behaviors form a bifurcation diagram with fractal structure

Feigenbaum constant

In a period doubling cascade, successive bifurcations double the period.

The ratio of successive interval lengths converges to the universal Feigenbaum constant.

Mandelbrot set

Pattern formation processes

Although governed by deterministic laws, chaotic systems exhibit unpredictable long-term behavior that often reveals emergent patterns.

→ Hidden order in chaos.

Strange Attractors

Phase space

A phase space represents the possible states of a system:

• plots state variables against each other
• reveals trajectories and dynamic patterns without displaying time as a separate axis

→ Used to analyze attractors and system behavior.

Lissajous curve

Chua’s Circuit

A nonlinear electronic circuit invented by Dr. Leon Chua (1983) that exhibits chaotic behavior.

It includes capacitors, an inductor, and Chua’s diode, which enables chaos.

The nonlinear Chua’s diode (N_R) on the right acts as an active resistor with a piecewise-linear – characteristic, essential for chaos. The inductor (L) on the left can be replaced by an active gyrator.

The system equations

The system is defined by three differential equations:

The differential equations describe the voltages and across capacitors and , respectively, and the current through the inductor , all as functions of time.

With the nonlinear function (breakpoints at ):

Where:

• lies parallel to : ,
• are the breakpoint voltages of the diode
• and are the slopes of inside and outside , both conductances

Double-scroll attractor

Normalization

The equations still carry physical units: and in volts, in ohms, and in farads, in henries.

Dividing all voltages by the breakpoint and rescaling time makes them pure numbers:

The dimensionless system

→ Turning the potentiometer changes , and with it .

Software implementation

Euler’s method approximates the next state from the current state and its rate of change:

• , , : the three state variables at step
• : the diode’s nonlinear – characteristic
• : how much of passes per step

Chua oscillator in Pure Data

fexpr~ approximates the three coupled differential equations in discrete time, reading its own previous output samples.

$y1[-1], $y2[-1], $y3[-1] are , , ; the signal inlets $x1, $x2, $x3 carry , , and can change while it runs.

fexpr~ $y1[-1] + $x1*$x2*($y2[-1] - $y1[-1] - (-0.71428571*$y1[-1]
       - 0.21428571*(abs($y1[-1]+1) - abs($y1[-1]-1))));
       $y2[-1] + $x1*($y1[-1] - $y2[-1] + $y3[-1]);
       $y3[-1] - $x1*$x3*$y2[-1]

→ The initial state must differ from zero: is an equilibrium.

Chaos in musical applications

• Used in modular synthesis and generative music
• Bridges randomness and structure
• Enables controlled unpredictability
• Ever-shifting sonic textures

Also: Chaotic FM, Chaos and S&H, etc.

Modular analog synthesis

Ian Fritz: Chaos modules for Eurorack systems

• Provide evolving, unpredictable modulation (different than random noise.)
• Enable multi-dimensional control across several synth parameters simultaneously.
• Small tweaks yield big changes.

Further examples

• Barry Truax: Chaotic processes in digital synthesis
• Dan Slater: Chaotic synthesis techniques for new timbres

Chaotic oscillators in electronics

• Rössler Oscillator
• Duffing Oscillator
• Van der Pol Oscillator

Literature

• Antoniou, Andreas. “Realisation of Gyrators Using Operational Amplifiers, and Their Use in RC-Active-Network Synthesis.” Proceedings of the Institution of Electrical Engineers, vol. 116, no. 11, 1969, pp. 1838–1850, https://doi.org/10.1049/piee.1969.0339.

• Choi, Insook. “A Chaotic Oscillator as a Musical Signal Generator in an Interactive Performance System.” Journal of New Music Research, vol. 26, no. 1, 1997, pp. 17–47, https://doi.org/10.1080/09298219708570715.

• Chua, Leon O. “Chua Circuit.” Scholarpedia, vol. 2, no. 10, 2007, p. 1488, https://doi.org/10.4249/scholarpedia.1488.

• Kennedy, Michael Peter. “Three Steps to Chaos—Part II: A Chua’s Circuit Primer.” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 40, no. 10, 1993, pp. 657–674, https://doi.org/10.1109/81.246141.

Literature (continued)

• Keuninckx, Lars, et al. “Simple Two-Transistor Single-Supply Resistor–Capacitor Chaotic Oscillator.” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 62, no. 9, 2015, pp. 891–895, https://doi.org/10.1109/tcsii.2015.2435211.

• Matsumoto, Takashi. “A Chaotic Attractor from Chua’s Circuit.” IEEE Transactions on Circuits and Systems, vol. 31, no. 12, 1984, pp. 1055–1058, https://doi.org/10.1109/TCS.1984.1085459.

• Mayer-Kress, Gottfried, et al. “Musical Signals from Chua’s Circuit.” IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, vol. 40, no. 10, 1993, pp. 688–695, https://doi.org/10.1109/82.246172.

• Mayer-Kress, Gottfried, et al. Musical Structures in Data from Chaotic Attractors. Beckman Institute, Centre for Complex Systems Research, University of Illinois at Urbana-Champaign, 1992.

Literature (continued)

• Slater, Dan. “Chaotic Sound Synthesis.” Computer Music Journal, vol. 22, no. 2, 1998, pp. 12–19, https://doi.org/10.2307/3680960.

• Szemplińska-Stupnicka, Wanda. Chaos, Bifurcations and Fractals around Us: A Brief Introduction. World Scientific, 2003.

• Truax, Barry. “Chaotic Non-Linear Systems and Digital Synthesis: An Exploratory Study.” Proceedings of the International Computer Music Conference (ICMC), 1990, pp. 100–103.

Websites

Schematics of chaotic circuits – Includes analog implementations and schematics of Lorenz, Rössler, and Sprott oscillators:

• Kleinschmidt, Glen. “Glen’s Stuff.” Glen’s Stuff, glensstuff.com/. Accessed 14 June 2025.

Provides schematics for classic chaotic systems such as Chua’s circuit, Lorenz attractor, and Rössler oscillator:

• “Chaotic Circuits.” CHAOTIC CIRCUITS, www.chaotic-circuits.com/. Accessed 14 June 2025.

Websites (continued)

Provides schematics and theory for chaos in synthesis such as driven double-well, three‑integrator chaotic attractors, with analog circuit diagrams suitable for modular/synth builds:

• Fritz, Ian. “Chaos Theory for Synthesizers.” The Electronic Sound-House, ijfritz.byethost4.com/Chaos/ch_over.htm. Accessed 14 June 2025.