Help & Reference
How to use Phase Evolution
Practical guidance for teachers, students and curious learners. Everything you need to navigate the application, follow the lessons, and get value from the interactive demonstrations.
Getting Started
Phase Evolution is a self-guided educational tool. No account is required — open a lesson, read at your own pace, and return to the Playground whenever a concept clicks.
We recommend starting with Lesson 1, "What Is Phase?", even if you are already familiar with oscillations. It introduces the vocabulary the rest of the series relies on.
Every page works on both desktop and mobile. Toggle light/dark mode from the button in the top navigation.
Lessons
Each lesson follows the same structure.
Lessons open with a plain-language overview, list clear learning objectives, present the key ideas, and provide an interactive demonstration. All five lessons are available now. Each lesson has a "Mark as complete" button that stores your progress locally in this browser.
Lesson 1: What Is Phase?
A quick guide to every part of the first interactive lesson.
What the phase circle represents
The circle is one complete cycle. The moving point shows where the system currently is within that cycle. Going all the way around the circle once corresponds to one full cycle — 360 degrees, or 2π radians.
Phase, ordering, and measured time
Phase evolves through an underlying relational ordering. The ordering parameter λ labels the succession of relational states via Φ(λ) = Φ₀ + Ω λ, independently of any clock. Once persistent structures (from Phase-Snap) provide stable reference processes, observers can assign numerical values to that ordering — measured time t. Phase evolution can then be written as Φ(t) = Φ₀ + ω t, an effective post-Phase-Snap description. Clocks measure the ordering of persistent physical processes; they do not create time.
Play and Pause
Play starts the phase evolving smoothly around the circle. Pause stops it exactly where it is so you can inspect the current phase.
Phase speed
The speed slider controls how quickly phase advances, measured in degrees per second. Speed changes only how fast the cycle is traversed — it does not change the shape of the cycle.
Direction
Clockwise and anticlockwise reverse the order in which phase positions are visited. The cycle itself is identical; only the order changes.
Manual phase slider
Drag the manual slider to set the phase directly. It is most useful when the simulation is paused, so you can move step by step through the cycle.
Why 0° and 360° are the same phase
A full cycle brings the system back to its starting position. So 0° and 360° describe the exact same point on the circle — they are two names for the same place.
What the changing colour means
The point's colour shifts continuously as it moves around the cycle. It is a visual cue only — every piece of information is also shown by the point's position, the labels around the circle, and the numeric readout.
How to reset
The Reset button returns everything to the starting state: phase back to 0°, direction clockwise, speed at its default value, and playback paused.
Lesson 2: Coupled Systems
A guide to every part of the second interactive lesson, where two phase systems influence each other.
What coupling means
Coupling is any connection that lets one repeating system influence the evolution of another. In the lesson, the strength of that influence is set by a single slider.
What the connecting line represents
The line between the two phase circles shows the coupling itself. It becomes thicker and more solid as coupling grows, and stays thin and dashed when coupling is weak or off. The line is a visual reminder — the influence is actually applied to how each system evolves.
How coupling strength works
With no coupling, the two systems evolve completely independently. Weak coupling produces only a small nudge — natural speeds still dominate. Medium coupling gradually pulls the pair toward a stable phase relationship. Strong coupling can lock them together.
Why the systems may drift
Each system has its own natural phase speed. When coupling is absent or too weak to overcome that difference, the phase difference between them keeps changing — they drift.
What phase difference means
Phase difference is the shortest angular gap between the two systems, measured between 0° and 180°. A small value means they are close together in their cycles; a large value means they are far apart.
What phase locking means
When the phase difference stops changing and stays within a small range for a short continuous period, the systems are described as phase locked. They still evolve, but together, at the same rate.
How to change natural speeds
Use the System A speed and System B speed sliders. Making the speeds different is what creates a tendency to drift when coupling is weak.
How to set the initial phase difference
The initial phase difference slider sets how far apart the two systems start. Changing it pauses the simulation first so nothing appears to jump.
What the disturbance button does
Apply Small Disturbance briefly shifts System B by a small phase amount. The coupling then determines whether the pair returns to its previous relationship or continues to drift apart.
Why stable does not always mean exact alignment
Two systems can be locked into a fixed but non-zero phase difference. That stable relationship is just as important as full alignment — it is the sign of a genuine, ongoing connection.
Lesson 3: Synchronisation and Coherence
A guide to every part of the third interactive lesson, where many phase systems organise into collective behaviour.
What each phase node represents
Every small circle is one independent phase system, complete with its own rotating marker, centre-to-marker line and natural phase speed.
What the connections represent
The faint lines between neighbouring nodes indicate that the systems can influence one another. The underlying model applies a shared coupling to all systems; the lines are a simplified visual reminder.
What synchronisation means
Synchronisation is when many separate phase systems settle into a shared phase relationship and continue evolving together — without any central controller.
What coherence means
Coherence measures how closely the individual phases are aligned. Low coherence means the phases are widely scattered; high coherence means they point in nearly the same direction.
How the coherence meter works
The meter shows coherence as a percentage from 0% to 100%, with a written interpretation beside it. It updates live as the simulation evolves.
Why natural speed variation matters
Systems with very similar natural speeds synchronise more easily. Systems with widely different speeds resist synchronisation and need stronger coupling to organise.
Why some systems synchronise faster than others
Stronger coupling and less speed variation both accelerate the approach to a coherent state. Fewer systems can also organise more quickly.
What partial synchronisation means
Sometimes many systems align while a few remain out of step. Coherence sits between low and high, and the collective phase is still meaningful.
The central collective phase indicator
The larger circle at the centre shows the mean phase direction of the whole group. It becomes more defined as coherence rises. When coherence is very low the mean phase is not meaningful, and the indicator shows "No clear direction".
What Randomise Phases does
Randomise Phases assigns new deterministic starting phases using a repeatable seed system, pauses the simulation and recalculates coherence. Pressing it again advances to the next seed.
What Disturb the System does
Disturb shifts a small group of nodes away from their current phases and shows a brief pulse around them. Watch whether the coupling restores coherence or the group continues to drift.
Why the visual layout does not change the model
Ring, Grid and Cluster only rearrange the nodes on screen. The underlying coupling model is unchanged, so the same coupling and speed variation produce the same coherence regardless of layout.
Why high coherence is not exact alignment
Even at very high coherence the individual phases are usually close together rather than perfectly identical. Coherence measures alignment as a group property, not point-by-point equality.
Lesson 4: Phase Waves
A guide to every part of the fourth interactive lesson, where disturbances travel through a chain of connected phase systems.
What the chain represents
Each circle is one phase system, and neighbouring systems are connected. A change to one system influences its immediate neighbours — and those neighbours in turn influence their neighbours.
Wave propagation
Clicking a node applies a small phase disturbance to just that node. Because the systems are connected, the disturbance does not stay put — it spreads to neighbouring systems as a travelling wave.
Coupling
Coupling sets how strongly each system feels its neighbours. Stronger coupling allows a disturbance to travel more rapidly along the chain.
Wave speed
The wave-speed control adjusts how quickly the disturbance moves through the chain. It works together with coupling — both raise the effective speed of propagation.
Damping
Damping gradually removes motion. With no damping a wave keeps bouncing around; with heavy damping the wave dies out quickly.
Reflection and boundaries
Fixed ends hold the outer nodes still, so waves bounce back. Free ends let the outer nodes swing, changing the reflected shape. Ring joins the two ends so waves keep circulating without a boundary.
Disturbance size and auto repeat
The disturbance size slider controls how strong each click is. Auto repeat sends a new disturbance from the first node at regular intervals so you can watch a steady stream of waves.
Energy indicator
The energy indicator is a simple visual cue: it rises when a fresh disturbance is added and falls as damping absorbs the motion or the wave spreads out over the chain.
Display options
Show trails leaves a short after-image of the phase markers. Show phase colours ties each marker's colour to its phase. Show wavefront highlights the node with the largest motion. Show node labels adds a small number under each node.
Lesson 5: Critical Transitions
A guide to every part of the fifth interactive lesson, where a group of connected systems crosses a threshold and reorganises.
What the system shows
The lesson reuses the many-node arrangement from Lesson 3. Sixteen coupled systems begin in a coherent state — a stable arrangement held together by coupling.
Applied energy
The Applied energy slider gradually widens the differences between the systems' natural speeds. At low values the group remains stable. As you raise it, strain builds until the arrangement can no longer hold.
Stage indicator
The stage indicator moves through Stable, Under strain, Reorganising, Reorganised and New stable arrangementas the system approaches and crosses the critical point.
Coupling and damping
Coupling determines how strongly systems hold each other in step, and damping smooths their response. Together they set how easily the group finds a new stable arrangement after crossing the threshold.
Apply small disturbance
Applies a small phase shift to a group of nodes. Near the threshold, a small disturbance can be enough to tip the whole system into a different arrangement.
Optional PDT panel
A separate expandable panel presents an interpretation proposed within Phase Differential Theory. It is clearly labelled and kept apart from the established educational content of the lesson.
Lesson 6: Interference
A guide to the two-wave interference visualisation.
Purpose: show that combining waves is addition and that the combined result depends on the relative phase and the two amplitudes.
Visualisation: three coloured traces on a shared horizontal axis — Wave A, Wave B and the combined wave.
Controls: Play/Pause and Reset. Sliders for relative phase (with 0°/90°/180°/270° quick buttons), the two amplitudes and the frequency. A direction toggle for opposite-travelling waves. Switches to hide either source or the combined wave.
Indicators: the legend shows the current amplitudes and the combined peak, plus the ΔΦ value.
What to observe: full reinforcement at ΔΦ = 0° and (for equal amplitudes) full cancellation at 180°. Different amplitudes prevent full cancellation.
Optional equation: y = A₁ sin(kx − ωt) + A₂ sin(kx − ωt + ΔΦ).
Limitations: a stylised one-dimensional model; not a physical wave-optics simulator.
Keyboard & reduced motion: every slider and button is keyboard accessible. Reduced motion pauses the animated pattern; the shape is fully readable from the static combined wave.
Lesson 7: Resonance
A guide to the driven-oscillator lesson.
Purpose: show that a system responds most strongly when driven near its natural frequency.
Visualisation: a mass on a spring driven by an external force arrow. A response chart appears after a frequency sweep.
Controls: Play/Pause, Reset, Apply pulse, Sweep frequency. Sliders for driving frequency, natural frequency, driving strength and damping.
Frequency Sweep: automatically changes the driving frequency while keeping the displayed slider, numerical label and visualisation synchronised. Starting a sweep also resumes playback; pausing the simulation pauses the sweep; Reset stops the sweep and restores the default frequency.
Indicators: live amplitude estimate, phase lag (degrees) and Δfreq. Response chart plots amplitude vs driving frequency after a sweep.
What to observe: the amplitude peak occurs near the natural frequency; the phase lag transitions through 90° across resonance; damping limits the peak.
Optional equation: ẍ + 2γẋ + ω₀²x = F cos(ωt).
Limitations: a single-degree-of-freedom model; real resonators have multiple modes and finite compliance.
Keyboard & reduced motion: all controls keyboard-accessible; reduced motion pauses the animation and lets the static amplitude/lag readouts convey the state.
Lesson 8: Phase Defects and Boundaries
A guide to ring and grid phase fields.
Purpose: show that a system can be locally ordered while carrying a global mismatch — a persistent defect.
Visualisation: ring mode places phases around a circle; grid mode shows a 2-D phase field with optional arrows and colours.
Controls: Play/Pause, Reset, layout (ring/grid), boundary mode (fixed/free/periodic), coupling, damping. Insert defect, Remove, Randomise. In grid mode, click to insert a defect; hold Shift + click for the opposite charge.
Damping: reduces the rate at which phases and defects move or relax. High damping does not erase a topological winding — it only slows change.
Indicators: winding number W (ring), and visible colour / arrow field.
What to observe: a full winding on a ring cannot be relaxed by local averaging. Opposite defects attract and cancel. Boundaries change what mismatches can persist.
Optional equation: W = (1/2π) ∮ dΦ.
Limitations: a simplified discrete phase model; real topological defects also involve amplitude structure.
Keyboard & reduced motion: keyboard-driven controls; reduced motion pauses continuous update and lets you read the current field.
Lesson 9: Competing Stable States
A guide to the double-well landscape.
Purpose: show that the same rules can lead to different stable outcomes and that history matters.
Visualisation: a curve V(x) with a ball resting in one of two valleys. The external bias tilts the landscape.
Controls: Play/Pause, Reset, sliders for bias, damping and noise. Buttons: Increase bias, Decrease bias, Kick system, Explore hysteresis (guided).
Indicators: current state (A, B, or between); recorded switching biases when the guided experiment runs.
What to observe: small kicks fail to switch; large enough kicks cross the barrier; the up-switch and down-switch biases differ (hysteresis).
Optional equation: V(x) = ax⁴ − bx² − hx.
Limitations: a one-dimensional caricature of bistability; real bistable systems have richer structure.
Keyboard & reduced motion: full keyboard control; reduced motion still lets you drive the guided experiment.
Lesson 10: Information in Phase
A guide to the sender/receiver capstone.
Purpose: show how bits can be encoded as phase and transmitted through a chain, subject to noise.
Visualisation: a chain of phase nodes from sender to receiver, with the original and decoded bit strings shown below.
Controls: a short binary message (max 8 bits) or three preset messages; Send, Pause, Reset. Sliders for transmission speed, coupling, damping, noise and receiver sensitivity. New noise pattern re-seeds the deterministic noise generator.
Damping: reduces transmitted signal strength as it moves through the chain. With no damping the signal amplitude is preserved apart from noise; with high damping the receiver sees a weaker, less reliable phase signal. Damping attenuates the signal — it is not simply noise.
Receiver sensitivity: controls how weak a received signal can be before decoding becomes unreliable. Low sensitivity requires a stronger incoming signal; high sensitivity allows weaker signals to be decoded. When the incoming signal is below the sensitivity threshold, the receiver marks that bit as uncertain (?) rather than silently guessing 0 or 1.
Indicators: bit errors count, accuracy percentage, seed value.
What to observe: stronger coupling preserves phase, noise creates errors, and the seed makes every run reproducible.
Optional equation: 0 → Φ = 0, 1 → Φ = π, with a nearest-reference decision rule.
Limitations: an educational model, not a real communications system. Real receivers use carrier synchronisation, filtering and error correction.
Keyboard & reduced motion: keyboard-accessible; reduced motion keeps the visual state legible while you control transmission speed manually.
Playground
The Playground is your laboratory.
The Playground offers nine modes, one per interactive lesson (single phase, two coupled, many systems, phase waves, interference, resonance, phase defects, stable states and phase information). Switch between them using the tabs at the top of the page. Each mode gets a fresh simulation, so switching cancels the previous animation loop and clears its state.
Controls
Every interactive control has a visible label and works with the keyboard.
- Play / Pause — starts or stops the phase evolution.
- Reset — restores the starting state of the current lesson.
- Phase speed — how quickly phase advances through the cycle.
- Direction — clockwise or anticlockwise.
- Manual phase — set the phase directly through one full cycle.
- Coupling — how strongly connected systems influence one another.
- Damping — gradually removes motion from the system.
- Applied energy (Lesson 5) — how far the group is being pushed away from its comfortable arrangement.
- Apply small disturbance — briefly shifts one or more systems to test the group's response.
Colour scale: markers use a smooth colour wheel keyed to phase (0° → 360°). Colour is a visual cue only — every number is also shown as text.
Settings
The Settings page collects everything that changes how the application looks and behaves.
- Theme — switch between light and dark mode at any time.
- Animation speed — scale decorative transitions between 0.5× and 1.5×.
- Reduce motion — shortens or removes non-essential motion.
- Show labels by default — turn on labels such as node numbers where lessons offer them.
- Show equations automatically — open the optional mathematics panel when a lesson loads.
- Reset progress — clear all lesson progress and achievements from this browser.
Progress & Achievements
Your progress is stored locally in this browser — no account, no server.
Each lesson is Not started, In progress or Completed. Opening a lesson marks it as In progress; pressing "Mark as complete" marks it as Completed and can unlock an achievement.
Achievements are lightweight badges that appear as toast notifications when unlocked. They include completing each lesson and finishing the whole series. You can view them at any time on the Settings page.
Accessibility
Phase Evolution is designed to be usable with a keyboard, a screen reader and reduced motion.
- Every interactive control has a visible label and an ARIA label.
- Sliders and buttons work with keyboard navigation.
- Live-updating status regions (stage, coherence, live explanations) announce changes to screen readers.
- Colour is never the only cue — every value is also shown as text.
- Reduce motion in Settings shortens transitions for users who prefer less animation.
- High-contrast dark mode is available at any time.
- Layouts adapt from mobile through tablet to desktop, and tap targets are sized for touch.
Frequently Asked Questions
Do I need a physics background?
No. Every lesson starts from everyday intuition. Mathematical notation, when used, is optional.
Can I use this material in my classroom?
Yes. Phase Evolution is designed for both self-study and classroom use.
Is my progress stored anywhere?
Only in this browser. Clearing site data or pressing Reset progress on the Settings page erases it.
Does the app claim to prove Phase Differential Theory?
No. Where PDT interpretations appear, they are labelled clearly and kept separate from established educational content.
Glossary
- Phase
- Where a repeating process is within its cycle. Usually expressed as an angle from 0° to 360°.
- Cycle
- One complete repetition of the process — starting position back to starting position.
- Phase speed
- How quickly phase advances through the cycle, measured in degrees per second (or radians per second).
- Radians
- An angle unit where one full cycle is 2π. Equivalent to degrees: 2π rad = 360°.
- Degrees
- An angle unit where one full cycle is 360°.
- Direction
- The order in which phase positions are visited — clockwise or anticlockwise around the cycle.
- Coupling
- A connection that lets two repeating systems influence each other's phase. The strength of the connection determines how large that influence is.
- Natural phase speed
- The phase speed a system would follow on its own, without any coupling to another system.
- Phase difference
- The shortest angular gap between two phase systems, measured from 0° to 180°.
- Phase locking
- When two coupled systems settle into a phase difference that stays within a small range for a sustained period. They continue to evolve together at the same rate.
- Disturbance
- A brief shift applied to one system to test how the coupled pair responds — whether it recovers or continues to drift.
- Stable relationship
- A state where the phase difference between two coupled systems barely changes, even if that difference is not zero.
- Synchronisation
- When many coupled oscillators settle into a shared phase relationship and continue evolving together.
- Coherence
- How closely the phases of many systems are aligned. Displayed as a percentage from 0% (widely spread) to 100% (strongly aligned).
- Collective behaviour
- Behaviour that belongs to the whole group of systems rather than any single one — for example, moving together.
- Mean phase
- The average phase direction of a group of phase systems. Only meaningful when coherence is not too low.
- Natural frequency variation
- How different the natural phase speeds of the systems are from one another.
- Partial synchronisation
- A state where much of the group is aligned but some systems remain out of step.
- Order parameter
- A single number that summarises how organised the group is. Here it is the coherence value R.
- Recovery
- The return of a coupled group toward coherence after a disturbance.
- Distributed system
- A group of connected systems that produces collective behaviour without any central controller.
- Wave propagation
- The way a disturbance travels from one system to its neighbours, and from those to their neighbours, forming a moving pattern.
- Damping
- A gradual loss of motion. In a chain, damping slows a travelling wave and eventually stops it.
- Reflection
- The behaviour of a wave that reaches a fixed or free boundary and turns back along the chain.
- Boundary
- The edges of the chain. Fixed ends hold the outer nodes still, free ends let them swing, and a ring joins the two ends together.
- Wave speed
- How quickly a disturbance travels along the chain. It increases with stronger coupling.
- Critical transition
- A point at which a small change in a control (such as applied energy) is enough to reorganise the entire system into a different stable arrangement.
- Threshold
- The particular value of a control at which a critical transition occurs.
- Oscillation
- A repeating back-and-forth or round-and-round motion that returns to its starting configuration.
- Frequency
- How many complete cycles occur in one second, measured in hertz. It is related to phase speed by ω = 2πf.
- Amplitude
- The size of the oscillation — for example, how far a pendulum swings from its resting position.
- Deterministic
- Behaviour whose future is fixed by its current state and rules. The same starting conditions always produce the same result.
- Emergent behaviour
- Group-level behaviour that arises from many simple local interactions, without any central controller.
- Interference
- The combined result of two or more overlapping waves. Reinforcement (constructive) happens when peaks align; cancellation (destructive) happens when a peak meets a trough.
- Constructive interference
- The overlap of waves whose phases align, producing a larger combined wave.
- Destructive interference
- The overlap of waves whose phases are opposite, producing a smaller — sometimes zero — combined wave.
- Resonance
- The large response of a system when it is driven near its natural frequency.
- Natural frequency
- The frequency at which a system prefers to oscillate on its own.
- Phase lag
- How far behind (in phase) a driven system is compared to the driving force.
- Q factor
- A measure of how sharp a resonance is. Higher Q means a taller, narrower peak.
- Phase defect
- A location where a phase field cannot be smoothed away by local changes — a persistent knot in the pattern.
- Winding number
- How many complete turns of phase are accumulated when tracing a closed path through a phase field.
- Topological protection
- The property that certain defects cannot be removed by any small local rearrangement.
- Bistability
- A system that has two stable states for the same external conditions.
- Hysteresis
- The dependence of a system's state on its history — the switching point going up differs from the switching point coming down.
- Potential landscape
- A curve or surface whose shape determines which states a system prefers and how easily it moves between them.
- Noise
- Small unpredictable variations added to a signal or system.
- Binary phase-shift keying
- Encoding one bit per symbol by transmitting a carrier wave with one of two phase values (typically 0 and π).
- Signal-to-noise ratio
- How large a signal is compared to the noise around it. Higher values are easier to decode.