Visual theory primer

Concepts for Reading Chaotic Systems

The toolbox is built around a teaching goal: students should connect definitions with reproducible graphics. The dictionary gives names to the concepts; this page explains how to read the figures, what each diagnostic can answer, and what should not be concluded from a single plot.

The Rule

An equation or map tells the state how to move. Without this rule, there is no dynamical system.

The Start

The initial condition launches the trajectory. In multistable systems it can change the final destination.

The Figure

Each plot answers a specific question about geometry, destination, frequency, sensitivity, or parameter change.

The Evidence

Chaos is interpreted by combining diagnostics, not by trusting one attractive image.

What makes a system chaotic?

Chaotic System

Definition: A chaotic system is deterministic, nonlinear, bounded in the observed regime, and sensitive to initial conditions. Deterministic means that the same rule and the same initial condition reproduce the same trajectory. Sensitivity means that two extremely close initial conditions can separate measurably as time passes.

How to read the figure: The Lorenz plot is not random noise. The orbit follows fixed equations, but the flow stretches and folds nearby states. That geometry makes the trajectory switch irregularly between lobes while remaining inside a bounded region.

Watch for: Irregular appearance is not enough. A poor time step, too short a simulation, unremoved transient behavior, or an unreadable projection can also produce confusing figures.

Lorenz trajectory with state time series

What changes with time?

Dynamical System

Definition: A dynamical system is a rule for updating a state. In a continuous flow the rule is a differential equation, such as dx/dt = f(x,y,z). In a discrete map the rule advances by iterations, such as x(n+1)=F(x(n)).

How to read the figure: Every point in the figure is a state of the system. The curve records the order in which the system visits those states. The time-series panels show how each coordinate changes while the phase portrait shows the geometry of the same motion.

Watch for: Do not confuse the dimension of the drawing with the dimension of the system. A high-dimensional model may be shown through only two or three displayed coordinates.

Lorenz 3D trajectory with 2D phase portraits projected on coordinate planes

Where does the motion live?

Phase Diagrams and Trajectories

Definition: A phase diagram plots state variables against each other. In a 2D portrait it may show x versus y; in a 3D portrait it may show x-y-z. Time is not the main axis. The plot shows the geometry of motion in state space.

How to read the figure: The 3D figure shows the Lorenz trajectory and the 2D portraits projected onto the coordinate planes. Those projections help students see how the same motion looks in x-y, x-z, and y-z views without losing the 3D context.

Watch for: A 2D projection can create apparent crossings that do not exist in 3D. Projections are useful for reading structure, but they also hide information.

Lorenz x-y, x-z, and y-z phase portraits

Why inspect more than one projection?

Separate 2D Phase Portraits

Definition: A 2D phase portrait is a projection of the state-space trajectory onto two selected variables. It simplifies the geometry so local folds, loops, and symmetries are easier to inspect.

How to read the figure: The x-y, x-z, and y-z panels are three views of the same Lorenz simulation. A feature that looks dense in one panel may separate in another. This is why the toolbox exposes multiple projections before export.

Watch for: A projection is not a new simulation. It is the same trajectory viewed through fewer coordinates.

Where does the trajectory go after transients?

Attractor

Definition: An attractor is the set approached by a family of trajectories after initial transient behavior. It can be a fixed point, a limit cycle, a torus, a chaotic set, or another invariant structure.

How to read the figure: The classical Lorenz attractor is not a single closed loop. The orbit repeatedly visits two lobes in an irregular order while remaining bounded. The rotating animation makes the folded 3D structure easier to see.

Watch for: A two-lobed figure does not prove chaos by itself. It should be compared with time series, bifurcation behavior, spectra, and Lyapunov estimates.

What changes when a parameter changes?

Bifurcation Diagram

Definition: A bifurcation occurs when a small change in a control parameter changes the qualitative behavior of the system. Equilibria can lose stability, cycles can appear, periods can double, and chaotic bands can emerge.

How to read the figure: Each column corresponds to one parameter value. The retained points show long-term values after the transient has been removed. A single branch suggests simple behavior; split branches, dense bands, or scattered event values suggest more complex regimes.

Watch for: A sparse or tiny diagram is easy to misread. Increase the parameter samples, retained points, integration time, and figure size only after the interval has been located.

How can a continuous flow be reduced to comparable returns?

Poincare Section

Definition: A Poincare section records intersections of a continuous trajectory with a chosen surface, often using a fixed crossing direction. It turns a flow into a sequence of return points.

How to read the figure: The figure shows the Lorenz trajectory, the section plane, and the crossing points. A periodic orbit would produce a small number of repeated points. A chaotic orbit produces a richer return structure.

Watch for: The section depends on the plane and crossing direction. A poorly chosen plane can show too few points or hide the behavior you meant to study.

How can an oscillation be born from an equilibrium?

Hopf Bifurcation

Definition: A Hopf bifurcation occurs when an equilibrium changes stability and a small oscillation appears or disappears around it. In a supercritical Hopf bifurcation, a stable limit cycle appears after the critical parameter is crossed.

How to read the figure: The conceptual figure contrasts a stable focus with the new oscillatory state and plots how the oscillation amplitude grows. It is a local mechanism for the birth of periodic motion.

Watch for: Hopf does not mean chaos. It explains how oscillations can appear; chaos usually requires additional mechanisms such as period doubling, intermittency, crises, or global interactions.

Lorenz basin map with representative trajectories from different colored regions

Which destination belongs to each initial condition?

Basins of Attraction

Definition: A basin of attraction is the set of initial conditions that approach the same long-term destination. When several destinations coexist, phase space is partitioned into colored regions.

How to read the figure: Each pixel in the basin map is an initial condition. The color is a class label, not a physical variable. The accompanying trajectories show how starts chosen from different regions fall into different destinations, matching the dictionary-style explanation.

Watch for: A basin is not the same as a coexistence plot. A basin classifies many initial conditions; a coexistence plot compares a few selected starts.

Animation of two Lorenz coexisting trajectories generated together

Can the same system have more than one destination?

Coexisting Attractors

Definition: Coexistence means that the equations and parameters remain fixed, but different initial conditions approach different long-term destinations. This is multistability when those destinations are stable under small perturbations.

How to read the figure: The Lorenz coexistence example uses sigma=10, rho=24.4, beta=8/3. Two symmetric starts are integrated together and approach two different stable equilibria. The animation emphasizes that both trajectories are generated by the same equations at the same time.

Watch for: Coexistence does not always mean two chaotic attractors. In this Lorenz example the coexistence is between two stable fixed points.

Animation showing expansion and contraction of perturbations

How is sensitivity to initial conditions measured?

Lyapunov Exponents

Definition: Lyapunov exponents measure average rates of expansion or contraction of small perturbations. A positive exponent indicates that nearby trajectories separate exponentially on average.

How to read the figure: The perturbation animation shows one direction expanding and another contracting. In a 3D continuous chaotic flow, a typical spectrum has one positive exponent, one near-zero exponent along the flow direction, and one negative exponent.

Watch for: Finite-time estimates can drift. Interpret the sign and magnitude only after checking transient removal, integration time, step size, and convergence.

Normalized FFT spectrum for a Lorenz trajectory

Which frequencies appear in a time series?

FFT and Spectral Reading

Definition: The Fast Fourier Transform converts a time-domain signal into frequency content. It helps compare periodic, quasiperiodic, and irregular signals.

How to read the figure: A simple periodic orbit often gives sharp peaks. A chaotic trajectory can spread energy over broader bands. The observed variable, sampling step, retained time window, and normalization all affect the spectrum.

Watch for: FFT is a supporting diagnostic. It should not be used alone to classify chaos.

Synthetic Sprott-style example generated by the toolbox

How can compact codes generate candidate dynamics?

Sprott Codes and Generating Functions

Definition: A Sprott-style code is a compact recipe for a map or flow. The family symbol, dimension, order, and coefficient characters are decoded into a mathematical function: x(n+1)=F(x(n)) for maps or dx/dt=f(x) for flows.

How to read the figure: After decoding, the toolbox simulates the function, discards transients, chooses a projection, and applies visual controls such as color, alpha, point size, and hidden-coordinate coloring. The resulting figure is a candidate for further analysis.

Watch for: A visual candidate is not proof of chaos. It should be tested with longer simulations, different initial conditions, smaller steps for flows, and diagnostics such as Lyapunov exponents.

What Question Does Each Plot Answer?

A graph is easier to interpret when its question is explicit. This table keeps diagnostics separated: a basin is not a coexistence plot, FFT is not a proof of chaos, and a beautiful trajectory is not enough to classify a system.

Plot	Useful for answering	Should not be used to claim
2D/3D trajectory	Where the orbit moves and whether it remains bounded.	It does not prove chaos by itself.
Time series	Whether a variable converges, oscillates, diverges, or remains irregular.	It does not automatically separate transient and final behavior.
Bifurcation diagram	How retained long-term behavior changes across a parameter.	It does not replace a detailed simulation at one parameter value.
Poincare section	How an orbit returns to a chosen surface.	It is not independent of the section plane or crossing rule.
Hopf diagram	How a local oscillation appears or disappears near an equilibrium.	It does not imply chaos by itself.
Basin map	Which destination belongs to many initial conditions.	It does not show the full 3D attractor geometry.
Coexistence plot	Whether selected starts reach different destinations under the same parameters.	It does not show the full partition of phase space.
Lyapunov estimate	Whether perturbations expand or contract on average.	It should not be read without convergence checks.
Sprott Explorer render	What a decoded compact function produces under chosen simulation and visual settings.	It does not turn a quick visual filter into a formal proof.

Recommended Mini Lab

This route follows the dictionary's purpose: connect one definition with one graph and one concrete action in the toolbox.

Start with the classical Lorenz system: sigma=10, rho=28, beta=8/3, x0=(0.1,0.1,0.1). Compare the 3D trajectory with z(t).
Inspect the projected 2D portraits. Notice what each projection reveals and what it hides.
Reduce rho below the classical chaotic regime, then increase it again. Compare convergence, oscillation, and lobe switching.
Open the bifurcation guide and sweep rho. Read each column as retained behavior after the transient has been discarded.
Compare the bifurcation diagram with a Poincare section: one sweeps parameters; the other records returns on a surface.
Use the Hopf figure to separate the birth of oscillations from a claim of chaos.
Open basins near rho=24.4 and read the colors as destinations for many initial conditions.
Open coexistence and compare the two registered Lorenz starts. Keep parameters fixed and change only the initial condition.
In the Sprott Explorer, decode a generated example and identify whether the underlying object is a map or a flow before interpreting the image.
Finish with FFT and Lyapunov diagnostics to compare visual impressions with spectral and perturbation evidence.