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Coding Challenge: Lorenz Attractor Visualization in Processing

Added: 2026-08-06

Deep Dive

Basic understanding of ordinary differential equations (ODEs) and how they model rates of change mathematically.

50 - Introduction to Ordinary Differential Equations

50 - Introduction to Ordinary Differential Equations

244

Ordinary differential equations (ODEs) model rates of change in single-variable systems, such as speed and acceleration as derivatives of position. Key concepts include: (1) Order - the highest derivative's degree, (2) Degree - the highest power of the highest derivative, (3) Multiple notations (dy/dx, y', ẏ, f(x)). To verify solutions, substitute the function and its derivatives into the equation and confirm the left-hand side equals the right-hand side. For example, y = 2x + 3 + (1/2)e^(3x) solves d²y/dx² - 3(dy/dx) + 2y = 4x + e^(3x).

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Modeling Introduction to Ordinary Differential Equations

Modeling Introduction to Ordinary Differential Equations

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An ordinary differential equation (ODE) contains at least one derivative with exactly one independent variable (typically time T). ODEs are distinguished from partial differential equations (PDEs), which involve multiple independent variables. The order is determined by the highest derivative present. ODEs are classified as autonomous (rate depends only on Y) or non-autonomous (rate depends on both T and Y). The modeling process involves defining variables, stating assumptions, writing the equation, solving, and analyzing. For the skydiver model, dy/dt = -32T, solved by integration to y = -16t² + C. Initial conditions (Y(0) = 15,000) determine the constant C. The bathtub analogy illustrates ODEs: Y represents accumulated amount, T is time, and dy/dt is the rate of change. For water filling at 8 gallons per minute, dy/dt = 8, solving to y = 8T. For elk population growth with 13% annual birth rate, dy/dt = 0.13Y (autonomous). Newton's Law of Cooling states dy/dt = -k(T - T₀), where the negative sign ensures cooling when T > T₀ and heating when T < T₀.

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Basic concepts of Differential Equation. #majhibira General Elective Math 1st sem CHSE Odisha.

Basic concepts of Differential Equation. #majhibira General Elective Math 1st sem CHSE Odisha.

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An ordinary differential equation is a differential equation involving total derivative with respect to a single independent variable. The total derivative represents the rate of change of a function with respect to one variable. For example, dy/dx = mx is a first-order ODE, and d²y/dx² + 5dy/dx + mx = 0 is a second-order ODE.

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Introduction to Ordinary Differential Equations

Introduction to Ordinary Differential Equations

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An ordinary differential equation (ODE) is an equation relating a function to its derivatives with respect to one independent variable; key concepts include the order (highest derivative present), autonomous vs non-autonomous equations (presence of independent variable), initial value problems requiring as many conditions as the order, parameters as context-specific constants, and equilibrium solutions which are constant functions satisfying the equation. Examples include the exponential growth/decay equation dy/dt = Ky, the logistic model dP/dt = K(1-P/N)P, the pendulum equation θ'' + (g/L)sin(θ) = 0, and the Lotka-Volterra predator-prey equations.

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Introducción a las ecuaciones diferenciales ordinarias

Introducción a las ecuaciones diferenciales ordinarias

770

Ordinary differential equations (ODEs) are equations containing derivatives of dependent variables with respect to independent variables, used to model physical problems. Many ODEs lack analytical solutions, requiring numerical approximations. The three main goals in studying ODEs are: (1) discovering the equation that models a physical situation, (2) finding the solution (analytically or numerically), and (3) interpreting the results. The rate of change concept is fundamental, where the derivative represents how a variable changes with respect to another.

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Deep Dive

Concepts of numerical integration, specifically Euler's method for approximating continuous equations using discrete time steps in code.

Mamba and S4 Explained: Architecture, Parallel Scan, Kernel Fusion, Recurrent, Convolution, Math

Mamba and S4 Explained: Architecture, Parallel Scan, Kernel Fusion, Recurrent, Convolution, Math

58.2K

To compute state space models on digital computers, we discretize continuous-time systems. The Euler method approximates derivatives using small step sizes: h(t+Δ) ≈ h(t) + Δ·dh/dt. For the bunny population model (dh/dt = λh), this gives h(t+Δ) = h(t)(1 + λΔ). This approximation allows us to compute the state at discrete time steps rather than solving the continuous differential equation analytically.

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Differential equations, a tourist's guide | DE1

Differential equations, a tourist's guide | DE1

5.3M

Euler's method provides a basic numerical algorithm for solving differential equations: define initial conditions for all state variables, then iterate over small time steps, updating each variable by multiplying its rate of change (from the differential equation) by the time step size. For the pendulum, this means updating θ by θ' × Δt and updating θ' by θ''(θ, θ') × Δt at each step. More sophisticated numerical methods exist to better balance accuracy and computational efficiency.

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NUMERICAL METHOD|EULER'S METHOD |MATHEMATICS|PRADEEP GIRI SIR

NUMERICAL METHOD|EULER'S METHOD |MATHEMATICS|PRADEEP GIRI SIR

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Euler's Method is a numerical technique used to approximate the solution of differential equations by iteratively calculating y values using the formula y_n = y_{n-1} + h * f(x_{n-1}, y_{n-1}), where h is the step size calculated as (x_final - x_initial) / number_of_steps, and f(x,y) represents the derivative function. The method involves starting with initial conditions (x₀, y₀), then sequentially computing x values by adding h, and corresponding y values using the formula, until reaching the target x value.

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المعادلات التفاضلية عبر بايثون - أسلوب أويلر Euler method

المعادلات التفاضلية عبر بايثون - أسلوب أويلر Euler method

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The Python implementation requires: (1) Defining the function f(t, x) that represents the right-hand side of the differential equation. (2) Setting parameters: initial time t₀, initial value x₀, step size Δt, and final time tₙ. (3) Creating time array using numpy.linspace. (4) Initializing solution array with x₀. (5) Implementing the Euler loop: for each time step, update x using x_new = x_old + Δt·f(t_old, x_old).

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Euler's Method | MIT 18.03SC Differential Equations, Fall 2011

Euler's Method | MIT 18.03SC Differential Equations, Fall 2011

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Euler's Method is a fundamental numerical technique for approximating solutions to ordinary differential equations by using tangent line approximations at discrete intervals. The method works by iteratively computing the next value using the formula y_{n+1} = y_n + h * F(x_n, y_n), where h is the step size and F represents the right-hand side of the ODE. To determine whether the numerical approximation overestimates or underestimates the exact solution, one should examine the concavity (second derivative) of the solution at the initial point: if the solution is concave down (negative second derivative), Euler's Method will overestimate the exact solution, as demonstrated when solving y' = y² - xy with y(0) = -1 and step size 0.5, yielding y(1) ≈ -0.25 which overestimates the true solution.

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Deep Dive

Familiarity with the Processing programming environment (Java-based), including setup, draw loops, and basic 3D rendering concepts.

Coding a Game for Complete Beginners [ 1 hour tutorial ]

Coding a Game for Complete Beginners [ 1 hour tutorial ]

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Processing uses two main functions: `void setup()` runs once at the beginning of the program for initialization, while `void draw()` runs repeatedly (typically 60 times per second) to create animation. Everything outside these functions runs only once top-to-bottom, while everything inside draw runs continuously in a loop.

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Processing Tutorial 1 - Getting to know Processing

Processing Tutorial 1 - Getting to know Processing

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Processing is a Java-based programming language designed for creative coding, featuring a simple interface with two primary functions: setup() for one-time initialization (like setting canvas size and background) and draw() for continuous animation; the environment includes an auto-format feature that automatically organizes code for readability, making it accessible for beginners to create visual graphics and animations with minimal code compared to traditional programming languages.

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Corso di Coding - Usare Processing - #2

Corso di Coding - Usare Processing - #2

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Processing is a beginner-friendly Java-based programming environment designed for creating visual graphics, animations, and interactive programs; it features a simple IDE with a text editor, toolbar for running/stopping code, and a console for output, where programs typically follow a basic structure with setup() for initialization and draw() for continuous drawing operations, though both functions are optional.

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Loops & stuff in C#. Also a whirldwind tour of Processing

Loops & stuff in C#. Also a whirldwind tour of Processing

121

This segment introduces Processing, a Java-based programming environment excellent for learning programming concepts quickly. The instructor explains the coordinate system with origin at top-left, the setup() method for initialization (runs once) and draw() method for animation (runs 60 times per second). The segment covers background() and stroke() methods for colors, including single-value gray shades and RGB color mixing. The instructor demonstrates using mouseX and mouseY for interactive graphics, and explains how to create animations by updating variables in the draw() method. The segment emphasizes that Processing allows learning loops, conditionals, and graphics in about one afternoon compared to much longer in C#.

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Coding Challenge #101: May the 4th Scrolling Text

Coding Challenge #101: May the 4th Scrolling Text

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Processing is a Java-based programming environment suitable for creating visual art and animations. To create a 3D project, you need to add the P3D renderer. The basic setup involves creating a window with specified dimensions (e.g., 800x600 pixels), setting the background color (e.g., background(0) for black), and defining the draw loop where the animation will occur.

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Deep Dive

Fundamental knowledge of 3D coordinate systems (X, Y, Z axes) and vector manipulation.

41) المتجهات في الفضاء - وحدة المتجهات - توجيهي المسار الأكاديمي

41) المتجهات في الفضاء - وحدة المتجهات - توجيهي المسار الأكاديمي

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A three-dimensional coordinate system extends the two-dimensional system by adding a Z axis perpendicular to both X and Y axes. This system uses three mutually perpendicular axes (X, Y, Z) to represent points in space with three coordinates (x, y, z). The three coordinate planes (XY, XZ, YZ) divide space into eight regions called octants. To plot a point (x, y, z): first plot (x, y) on the XY plane, then draw a line parallel to the Z axis to reach the correct Z coordinate. A vector in three-dimensional space has three components corresponding to the X, Y, and Z axes. If a vector starts at point A(x₁, y₁, z₁) and ends at point B(x₂, y₂, z₂), its components are: (x₂ - x₁, y₂ - y₁, z₂ - z₁). The magnitude of a vector v = (v₁, v₂, v₃) is |v| = √(v₁² + v₂² + v₃²). Vector operations include: addition (add corresponding components), subtraction (subtract corresponding components), and scalar multiplication (multiply all components by the scalar).

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VECTOR ALGEBRA II Part -1II BY JP SIR II

VECTOR ALGEBRA II Part -1II BY JP SIR II

229

Vector Algebra is an important topic in Class 12th Mathematics and Class 11th Physics that improves 3D Geometry understanding. The 3D coordinate system consists of three mutually perpendicular axes: X-axis (horizontal), Y-axis (vertical), and Z-axis (coming out of page). Each axis has positive and negative directions. Unit vectors î, ĵ, and k̂ represent the positive directions of X, Y, and Z axes respectively. When referring to individual axes, we say X-axis, Y-axis, or Z-axis; when referring to all three together, we call it the coordinate system.

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الصف الثالث الثانوي/مراجعة الوحدة الاولي هندسة فراغية 2026🔥/اقوي تلخيص وحل/مع المهندس احمد سامي سالم

الصف الثالث الثانوي/مراجعة الوحدة الاولي هندسة فراغية 2026🔥/اقوي تلخيص وحل/مع المهندس احمد سامي سالم

173

The three-dimensional coordinate system consists of three mutually perpendicular axes: x-axis (محور السينات), y-axis (محور الصادات), and z-axis (محور العين). These axes are pairwise perpendicular, meaning any two axes form a right angle. The system includes three coordinate planes: xy-plane (المستوى س ص), yz-plane (المستوى ص ع), and xz-plane (المستوى س ع). When two planes intersect, they intersect along one coordinate axis. When all three planes intersect, they meet at the origin (0,0,0). Points on axes have two zero coordinates; points on planes have one zero coordinate. The distance between two points A(x₁,y₁,z₁) and B(x₂,y₂,z₂) is d = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]. The midpoint is ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2). The centroid of a triangle with vertices A, B, C is ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3, (z₁+z₂+z₃)/3). A vector is a directed line segment with a starting point, ending point, and direction. Two vectors are equal if they have the same magnitude and direction. Any vector can be expressed as endpoint minus starting point. Vector addition uses two geometric rules: the triangle rule (place tail of second vector at head of first, sum is vector from first tail to second head) and the parallelogram rule (place both vectors from same origin, sum is diagonal). Vector subtraction places both vectors from same origin, difference is vector from endpoint of second to endpoint of first.

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3d Geometry x, y and z axis की सम्पूर्ण जानकारी

3d Geometry x, y and z axis की सम्पूर्ण जानकारी

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3D geometry (tridimensional geometry) involves shapes and figures in three-dimensional space. The system consists of three perpendicular axes: X-axis, Y-axis, and Z-axis, which intersect at the origin. These axes form three coordinate planes: XY-plane, YZ-plane, and ZX-plane. A point in 3D space is represented by three coordinates (x, y, z) in the format (x, y, z), where each value represents the position along the respective axis. Points on individual axes have specific coordinate patterns: X-axis points are (x, 0, 0), Y-axis points are (0, y, 0), and Z-axis points are (0, 0, z).

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شرح عالمى لدرس | المتجهات في الفراغ | فرع الهندسة الفراغية الصف الثالث الثانوي

شرح عالمى لدرس | المتجهات في الفراغ | فرع الهندسة الفراغية الصف الثالث الثانوي

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This lesson covers the three-dimensional coordinate system with three mutually perpendicular axes (x, y, z) dividing space into eight octants. Key concepts include: (1) The three axes are mutually perpendicular (متعامدة مثنى مثنى); (2) The axes create three coordinate planes: xy-plane, yz-plane, and xz-plane; (3) A vector is a directed line segment with a starting point, ending point, and direction; (4) The position vector has its starting point at the origin (0,0,0) and ending point at P(x,y,z), written as (x,y,z); (5) A vector has three components: x-component, y-component, and z-component; (6) The magnitude (معيار) of a vector a = (x,y,z) is |a| = √(x² + y² + z²), the three-dimensional extension of the Pythagorean theorem.

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Core Concepts

Initial Coding

Debugging Issues

2D Plotting

3D Visualization

Visual Enhancements

Interactive Play

Creative Challenge

Core Concepts

0:00

Playing Section

1
Explains differential equations and the Lorenz system's mathematical foundation.
2
Describes chaotic behavior and the butterfly effect in nonlinear systems.

Deep Dive

Implementation of more precise numerical integration techniques, such as the 4th-order Runge-Kutta (RK4) method, to reduce approximation errors.

lec05 - Forward Euler, Runge-Kutta, local error, global error, solve_ivp()

lec05 - Forward Euler, Runge-Kutta, local error, global error, solve_ivp()

597

The fourth-order Runge-Kutta method uses four intermediate estimates to achieve high accuracy with reasonable computational effort. The scheme is: k₁ = h*f(uₙ, tₙ); k₂ = h*f(uₙ + k₁/2, tₙ + h/2); k₃ = h*f(uₙ + k₂/2, tₙ + h/2); k₄ = h*f(uₙ + k₃, tₙ + h); uₙ₊₁ = uₙ + (k₁ + 2k₂ + 2k₃ + k₄)/6. For u' = u, this gives uₙ₊₁ = (1 + h/6 + h/3 + h/3 + h/6)*uₙ = (1 + h)*uₙ, but with much smaller error per step.

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Programming for AI (AI504, Fall 2020), Class 14: Neural Ordinary Differential Equations

Programming for AI (AI504, Fall 2020), Class 14: Neural Ordinary Differential Equations

19.6K

The fourth-order Runge-Kutta method (RK4) provides significantly better precision than Euler's method by computing four intermediate slopes (k₁, k₂, k₃, k₄) at different points within each time step. k₁ uses the initial slope, k₂ uses a half-step prediction based on k₁, k₃ uses another half-step prediction based on k₂, and k₄ uses the full step prediction based on k₃. The final step is computed as y₁ ≈ y₀ + (k₁ + 2k₂ + 2k₃ + k₄)/6 × Δt. While RK4 is approximately four times slower than Euler's method due to its multiple evaluations, it produces much more accurate results.

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Resolución Numérica de un problema de valor inicial (EDO) por varios métodos

Resolución Numérica de un problema de valor inicial (EDO) por varios métodos

955

The Runge-Kutta 4th order (RK4) method is a higher-order numerical method for IVPs that uses four function evaluations per step. The implementation involves: (1) computing k1 = f(t(n), y(n)), (2) computing k2 = f(t(n)+h/2, y(n)+h*k1/2), (3) computing k3 = f(t(n)+h/2, y(n)+h*k2/2), (4) computing k4 = f(t(n)+h, y(n)+h*k3), (5) updating the solution as y(n+1) = y(n) + (h/6)(k1 + 2k2 + 2k3 + k4). This method is more accurate than Euler's and trapezoidal methods.

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Runge-Kutta Integrator Overview: All Purpose Numerical Integration of Differential Equations

Runge-Kutta Integrator Overview: All Purpose Numerical Integration of Differential Equations

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The RK4 method achieves fourth-order accuracy through a sophisticated combination of four intermediate vector field evaluations: F1 at the base point, F2 after a half-step in F1, F3 after a half-step in F2, and F4 after a full step in F3. The update uses a weighted average: x(k+1) = x(k) + (Δt/6)[F1 + 2F2 + 2F3 + F4]. This design cancels error terms up to O(Δt⁵) locally and achieves O(Δt⁴) global accuracy. The coefficients (1,2,2,1) reflect the relative trustworthiness of intermediate estimates, with half-step evaluations weighted more heavily. RK4 represents the standard workhorse algorithm for most numerical integration tasks.

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Métodos de Runge-Kutta: Euler Modificado, Heun y RK4 - Análisis Numérico

Métodos de Runge-Kutta: Euler Modificado, Heun y RK4 - Análisis Numérico

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This comprehensive section covers the RK4 (Runge-Kutta 4th order) method, the most widely used numerical method for solving differential equations. The RK4 method approximates the Taylor series of order 4, involving four function evaluations per step (k1, k2, k3, k4) combined with specific weights: w_new = w_old + h/6 * (k1 + 2*k2 + 2*k3 + k4). The method achieves fourth-order accuracy with error proportional to h^4, providing excellent precision for the same step size compared to lower-order methods. Implementation in Excel requires careful attention to evaluating the function at appropriate points for each k value. The method produces significantly smaller errors, achieving precision of four decimal places at most nodes, outperforming the RK3 method by approximately one decimal place. Errors accumulate over the interval, with the last node typically showing slightly reduced precision due to accumulated round-off errors.

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Deep Dive

Exploration and visualization of other famous chaotic systems, such as the Rössler attractor, the Aizawa attractor, or Clifford map fractals.

Tracks of Light, Attractors (Aizawa)

Tracks of Light, Attractors (Aizawa)

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The Aizawa attractor is a chaotic attractor that demonstrates complex, non-repeating patterns in dynamical systems, visualized through this animation created using glChAOSP software.

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Fractals & Strange Attractors: Chaos Geometry Explained

Fractals & Strange Attractors: Chaos Geometry Explained

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The Rössler attractor, discovered by Otto Rössler in the mid-1970s, is a simple model of chaos with only one quadratic nonlinearity (x*z). Its equations are x_dot = a - y - z, y_dot = x + a*y, z_dot = b + z*x - c*z. With parameters a = b = 0.2 and c = 5.7, it produces a single-winged attractor. Visualizing trajectories, an initial ellipsoid of initial conditions gets stretched, folded, and reinjected repeatedly. Cross-sections reveal gaps resembling the Cantor set construction. The attractor contains an infinite complex of surfaces that never truly merge due to uniqueness of solutions. This demonstrates how stretching, folding, and reinjection generate fractal microstructure in chaotic systems.

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Clifford attractor tutorial Max/MSP/Jitter

Clifford attractor tutorial Max/MSP/Jitter

1.3K

The Clifford attractor is a 2D strange attractor that can be implemented in Max/MSP/Jitter using matrix operations with trigonometric functions (sign, cosine, sine) applied to X and Y coordinates, where the resulting values are fed back into the system to generate the characteristic fractal pattern; the attractor requires two planes of noise for initial positioning and four parameters that control its behavior, with optimal results typically occurring when parameters are kept in the range of -1 to -2.

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week3 class Fall, 2023 -- "Sound: Advanced Topics I"

week3 class Fall, 2023 -- "Sound: Advanced Topics I"

108

Paul Burkhardt at the University of Sydney provides extensive resources for studying chaos and fractals, including equations for various attractors (Rossler, Volterra, Clifford), example code for running them, and the specific coefficient values needed for each equation. These resources help researchers and students understand and implement chaotic systems without having to derive everything from scratch.

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Rossler Attractor Pattern using Python

Rossler Attractor Pattern using Python

118

The Rössler Attractor is a famous chaotic system defined by three coupled differential equations: dx/dt = -y - z, dy/dt = x + Ay, and dz/dt = B + z(x - C). To visualize this attractor in Python, one must import numpy for numerical operations, matplotlib.pyplot for plotting, and solve_ivp from scipy to numerically solve the differential equations. The process involves defining the Rössler system function with parameters A, B, and C, setting initial conditions (typically x=0, y=1, z=0), defining a time interval and number of evaluation points, solving the ODE system using solve_ivp with appropriate accuracy settings, extracting the solution arrays, and creating a 3D plot to display the characteristic chaotic trajectory.

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Deep Dive

Constructing bifurcation diagrams to visualize and study how changing system parameters (such as the Rayleigh number) leads to stable, periodic, or chaotic behavior.

Explaining the Bifurcation Diagram for the Lorenz Chaotic System

Explaining the Bifurcation Diagram for the Lorenz Chaotic System

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Bifurcation diagrams visualize how system behavior changes as a single parameter varies, with the x-axis representing parameter values and the y-axis displaying Poincaré section intersection points. Construction involves: fixing all parameters except the studied one, choosing parameter ranges and iteration steps, simulating the system, discarding transient behavior, identifying intersection points, optionally reinitializing initial conditions, and plotting results. Single discrete points indicate periodic behavior (period one, two, etc.), while dense regions indicate chaos. The diagram reveals critical parameter thresholds where qualitative changes occur, providing immediate insight into system behavior across parameter space.

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Lecture 2 (updated): Sources & sinks, basins of attraction, bifurcations

Lecture 2 (updated): Sources & sinks, basins of attraction, bifurcations

2.7K

A bifurcation diagram shows the long-term steady-state behavior of a dynamical system as a parameter varies. To construct it, one iterates the function many times (discarding transient behavior) and plots only the final values approached. Stable fixed points appear as single points, stable periodic orbits appear as multiple points corresponding to their period, and chaotic behavior appears as continuous bands. Only attracting objects (sinks) appear in the diagram; repelling objects (sources) do not show up because they are never approached by random initial conditions.

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Nonlinear Dynamics: Constructing The Bifurcation Diagram

Nonlinear Dynamics: Constructing The Bifurcation Diagram

10.6K

A bifurcation diagram plots x_n versus R, with each vertical slice representing the attractor for that parameter value. To construct it, one removes the transient (initial transient behavior) from trajectories and plots only the asymptotic behavior. The diagram shows how attractors change as parameters vary, revealing bifurcations as branches splitting or merging. For R=2, all points collapse to a single fixed point; for R=2.5, two points appear (2-cycle); for higher R, more points emerge showing period-doubling cascades.

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Dynamical Systems And Chaos: The Bifurcation Diagram Part 2

Dynamical Systems And Chaos: The Bifurcation Diagram Part 2

5.5K

Bifurcation diagrams for the logistic equation reveal how small changes in the growth rate parameter r produce dramatic qualitative changes in system behavior, showing transitions from stable fixed points to periodic cycles through period-doubling bifurcations, and eventually to chaotic behavior with embedded periodic windows; constructing these diagrams requires specifying the number of initial iterates to ignore (transients), the number of iterates to plot, and iterating over many r values to capture the full range of dynamical behaviors.

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Dynamical Systems And Chaos: Hysteresis (Optional)

Dynamical Systems And Chaos: Hysteresis (Optional)

5.5K

Bifurcation diagrams visualize how system equilibria change with parameters. The construction process involves: (1) selecting parameter values, (2) plotting the function f(x) = Rx + x³ - x⁵, (3) identifying fixed points where f(x)=0, (4) classifying stability by crossing direction, and (5) connecting corresponding points across parameter values. The resulting diagram reveals complex structures including lines of stable (red) and unstable (blue) fixed points, showing how multiple attractors coexist for certain parameter ranges.

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Deep Dive

Leveraging GPU-accelerated particle systems and custom GLSL shaders to render and animate millions of points along the attractor path in real-time.

I Tried Rendering Millions Of Particles

I Tried Rendering Millions Of Particles

546.4K

This video explains how to implement an efficient GPU-based particle system for rendering iterated function system (IFS) fractals in real-time. The key insight is that while the formal mathematical method for generating IFS attractors requires exponential computational complexity (3^n particles for n generations), the chaos game method provides a linear-time stochastic alternative that works well with GPU parallelism. By combining GPU instancing for point cloud rendering with compute shaders for position updates, and using voxel-based ambient occlusion for lighting, the system can render hundreds of millions of particles efficiently. The video demonstrates that modern GPUs, particularly high-end models like the NVIDIA RTX 4090, can handle these workloads while achieving visually appealing results through techniques like linear interpolation smoothing and post-processing effects.

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Миллионы интерактивных реалтаймовых GPU-частиц на GLSL в Touchdesigner - полная версия

Миллионы интерактивных реалтаймовых GPU-частиц на GLSL в Touchdesigner - полная версия

1.9K

This tutorial demonstrates how to create massive real-time particle systems (up to several million particles) in Touchdesigner using GLSL shaders, covering camera control with spring physics, particle emission and lifespan management, texture-based positioning, and sound-responsive interactions, while addressing optimization techniques for high-performance rendering.

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Argos Vu – Peter Kenny, David Johnson

Argos Vu – Peter Kenny, David Johnson

172

Modern GPU compute shaders enable parallel processing of hundreds of thousands to millions of particles per frame, each running its own simple program. This technology creates charge point clouds where particles behave like electric charges seeking opposite polarity. The system supports platonic solids as geometric starting points for positioning attractors, with users able to teleport to any attractor for miniaturized perspective surveying. Velocity controls direction and magnitude, while trigger engagement enables flight. This technology serves as both a particle accelerator and precision point cloud modeling tool, with applications in 3D printing and intuitive vertex sculpting of platonic solids.

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OpenGL 4.5 勉強会 #24【最終回】「パーティクルシミュレーター」【プログラミング実況】

OpenGL 4.5 勉強会 #24【最終回】「パーティクルシミュレーター」【プログラミング実況】

859

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Core Concepts

Initial Coding

Debugging Issues

2D Plotting

3D Visualization

Visual Enhancements

Interactive Play

Creative Challenge

Core Concepts

0:00

Playing Section

1
Explains differential equations and the Lorenz system's mathematical foundation.
2
Describes chaotic behavior and the butterfly effect in nonlinear systems.

415.1K views8.9K21:21@TheCodingTrainOriginal Release: 2016-05-06

The Lorenz Attractor is a system of ordinary differential equations (dx/dt = σ(y-x), dy/dt = x(ρ-z)-y, dz/dt = xy-βz) that exhibits chaotic, nonlinear behavior where small changes in initial conditions produce vastly different outputs (the butterfly effect); this system can be visualized in 3D using Processing by implementing the equations with specific constants (σ=10, ρ=28, β=8/3), storing trajectory points in an array list, and drawing them with beginShape() and endShape() to connect the points.

[00:00:00]hello and welcome to another coding challenge in this coding challenge I am going to implement the system you see over here on the right called the lorenza tractor visualize that in 3D in processing all in one session here which will hopefully be about 10 to 20 minutes we'll see how long it takes I've got a timer running up there okay so before I start actually writing the code for it I want to talk about what this system is why does it exist why is it interesting and how does the math behind it work on a kind of um you know higher level and then we'll start looking at the formulas for it and implementing those formulas in code okay so the Loren system you know there's a lot of resources for this online I'm going to put a few in the description of this video Paul Bor site is a particularly useful one that I'll just mention really briefly but I'm going to look just at the Wikipedia page here which says the Lorent system is a system of ordinary differential equations okay so the first thing we have to tackle here is what is a differential equation I could easily I got a timer running here I could easily be here for like the rest of the day talking about differential equations and confusing myself over and over again but the basic idea of a differential equation is an equation that looks at how a certain variable changes according to other variables how the rate of change of a variable is affected by other variables in a system that rate of change being a differential so let me give you an actual practical real world example that might help solidify this understanding let's say you have a spring okay so I have a spring connected to a wall with a Bob at the bottom and you could think of this spring as having a position called y How does its y position change over time that's often written as the change in y over time you could think of that in programming speak as y equals y plus the change in y right every frame of Animation y changes but it changes based on it's uh the the rate of change of Y actually changes based on the Y position itself if you pull all the way down here how fast it's going to spring back is going to be much more than if you pull it over here so dy over DT the change in y the differential of Y is a function of other variables in the system itself y itself this is a differential equation and so you know this actually comes up all the time in programming you have position you have velocity velocity being the derivative the rate of change of position this is a something that you'll see over and over again and here the Loren system is defined by these particular differential equations the rate of change of X over time the rate of change of Y over time and the rate of change of Z over time now these variables X Y and Z those are those Define the state of the system but I'm going to use those literally as Xyz positions to visualize uh this particular to get this particular structure so if I look at those POs those those values the state XYZ over time and draw them with colors and lines I'm going to see this particular um this particular system so why is this system in interesting it turns out that the luren system which was um you know invented or discovered in 1963 by Edward Edward Edward Edward Edward Loren um you know he was studying meteorological conditions the flow of the atmosphere this type of thing it's one of these chaotic nonlinear systems meaning whatever the inputs you feed into the system uh um you feed some inputs into the system you calculate that system over time it's a deterministic system that will produce an exact result every time with the same inputs but if you change the input ever so slightly you could get a widely different output it's nonlinear a tiny change to the input makes a large change to the output this is the butterfly effect a little butterfly flaps its wings changing the air just a tiny bit in the uh outside on one side of the world and on the other side of World a giant volcano erupts okay so this is this idea of a nonlinear system it's chaotic it's unpredictable um and a lot of these types of systems exist in nature and this is one particular example of a model of that type of system okay so now let's look at these formula and let's try to figure out how we're going to actually make this happen in code I think we're ready so we know that I need three variables an x a y and a z so let me go to my uh program here and I'm going to close this one and I'm going to start start writing some code from scratch and I am going to give myself an X and A Y and a z and I'm going to give them some initial values uh like that okay so we're we're good so I solve those equations I know I need an x a y and a z so that's a good start now one thing I should mention really briefly as I'm four about five minutes into this is that notice I defined those values as floats if you look at examples you'll find on the internet often the the data type that will be used is a double a double being just like a floating Point number but having more decimal places and sometimes with these systems you really need a high degree of accuracy but I'm just some person here on the internet programming colors and things on the screen I don't care if it's a little bit not so accurate and Float is a bit more native to processing so I'm going to use floats um okay so now what I need to figure out is I want to kind of implement these equations so let's just start with x the change in X over time equals uh Sigma that's the Greek letter Sigma times y- x so first of all I I can't use the Greek letter Sigma easily in my code so Sigma I'm going to make a row I'm going to make B and beta I'm going to make C which I I realize is a bit confusing but that's what I'm going to do so those are constants by the way I need constants for those values so let me go back to my code and I'm going to say a float a equals 1 so I'm just going to start with the constant one now what I'm going to do is have this particular calculation happen over and over over time right it's supposed to happen over time time this is the interesting thing about this those formulas the change in X over time we're those formulas are talking about real life time that's continuous our time in processing is frame by frame by frame so every cycle through draw is one unit of time so what I'm going to say is that the change in x equals that constant time what y - x y - x and now X = X Plus that change in X it's really that simple right I'm just taking that exact formula that says the change in X over time equals the constant times y - x and now change in X just means change X by that value xal X Plus DX so I can actually start to go and Implement Y and Z the change in y and the change in Z and now I just need to go back and refer to those formulas again x * B minus zus y oh God I hope I get this right this is hard to do oh no sorry x x times B minus Z * y did I get this right minus y I want to have spaces there what's wrong here oh it doesn't know what B is I'll fix that later and now the change in Z is x * * y - c * Z so it's x * y - c * Z so this should be good and now I just need to get some spacing here and I need to also add uh two more constants float b equals 1 float Cals 1 and now I need to just change x y and z x y z x y z y z Dy DZ okay so this is the idea I Implement those formulas directly from the Wikipedia page to calculate the change in X the change in y the change is the and then actually change them now here's the thing what is my time there's a weird sort of thing that I'm going to bring up here which is my time step so right now every frame is like a singular unit of time but the the pattern that I'm going to draw right is going to look you know I'm not going to be able it's going to look something like this and in the sort of continuous world I was moving this marker continuously but in the processing animation world I have point point point point point so how close and I might connect those with a line so do I want to see my result look like this or do I want to see it as more continuous curve how fine do I want those to be and that's the sort of DT right if you think about oops I'm going to come back over here right really I'm saying DX over DT equals something so what I want to do is actually take that something something and multiply it by DT what is the change of time for each frame so what I'm going to do is also introduce a variable called DT and I'm going to say it's like 0 point I'm just going to make that one for right now but I'm going to take the whole thing and multiply that by DT for each one of these to sort of scale and I'll show you how different that's going to look um uh as we um as I continue doing this so now we've got those formulas implemented now we've got to see is anything working correctly here so I need a way of figuring that out the first way I'm going to do to figure that out is first just make sure I the program runs I'm going to add a black background and I'm going to run it and okay so I don't have any errors that's a good sign you know one thing I could do is I could say print line XYZ so I could sort of look at what's being printed out now notice something here all I'm doing is getting the values of zero so there's a little bit of an issue here which is that that doesn't make any sense so I can't start with 0 if I start with 00 these formulas are only ever going to produce 00 so I got to modify the inputs a little bit and I'll just start with X as like 0.01 that should do pretty well and if I run this again we should see look look I'm getting some interesting values there now what are these values producing anything of note not really but why not well because my constants I just made my constants one one and one so the Loren attractor Works a particular way based on what those input constants are so if I go back to the Wikipedia page I could sort of find um somewhere on here it's going to give me some values I could use like Sigma equal 10 rowal 28 and beta equal 8 / 3 so let's use those constants I believe Sigma was a b was um B was 28 and C was uh 8 divided 3 and I'm going to do that just to make sure I get a floating point value okay so I think that should should work I'm like skeptical and we should see o not a number okay so what have I done wrong here I might have to time out this video see how long this debugging takes and this might get edited out if it starts to take a really long time but let's look here um so what have I done wrong here uh XYZ um so let's let's comment out and just do X for a second uh and see what I get oops and I just want to look at X oh I had some numbers and then I got not a number for y so I might just be like one is a terrible number for DT because I'm like it's as if I'm teleporting myself Through Time massively fast so let me just make this 0.01 yeah okay this is more like what I expected to see Al those numbers are getting incredibly small I hope I implemented this right I have to check my formulas let's go back here and see if I'm getting some reasonable formulas okay good reason okay so this is better this is what I expected to see okay great so now I'm seeing some numbers that make more sense so my issue there was literally just having and think that won't get edited out was just having the number one it and it just sort of went off to Infinity super fast these systems um they don't have a recognizable repeating pattern and they tend towards they often tend towards infinity or some like non-existent value okay so let me go back to 0.01 now I want to be to figure out is something actually working here so the first thing I'm going to do is just say stroke 255 and I'm going to I'm also going to translate to the center of the window and I'm just going to draw a point and I'm just going to do this in 2D so I'm not going to worry about the Z I just want at least see like let me plot those the state of the system over time with XY so let's do that and ah good this is very good so it looks like I did the formulas correctly you can start to see like that's something there so I'm actually we're kind of like we're Trucking along here I've got the Loren system like sort of visualizing right now so that really basically I'm done however I want to make this look like something I want to make this look like a 3D shape I want to maybe add some color to it I want to be able to spin around it I want to I need to do more with it maybe I want to be able to like explode it so that all the points should like fly off of it and come back so I want to do even though in 13 minutes we kind of got the math down I want to do more with this in terms of the visual output of it so um so let's go forward with that so the first thing that I'll do is let's just go ahead and work in 3D so I'm going to change the renderer to p3d and then I'm going to add the Z point so we should see now that um it's the same it looks pretty similar but there's actually a z value there now one thing I want to add also is just I want to use the uh the processing uh function scale so scale is a transformation you can add to your scene to kind of like stretch it out a little bit might be to get this through zooming in and out as well but I just want all the points to be sort of spaced out a bit more and look a little bigger so I'm going to basically scale everything by five so we can see yeah so this looks so now I kind of have a bit of a closer view you can see all the points of Loren's tractor are being plotted in 3D now here's the thing how am I getting all these points on the screen I'm just drawing one point at a time well notice how my background is in setup so this worked just to see that the system was working but it's not going to be a very good solution for eventually over time being able to manipulate this system so what I need is some sort of data structure some sort of list that I can store all the points of the Lorenzo tractor so let me do that and what I'm going to do for that is an array list this is where we might rather be in JavaScript and I'll try to make a JavaScript version of this as well I'm going to call that points I'm going to make that a new empty array list so this is the syntax for something in Java called an array list which is a flexibly sized array something I could just add to and I'm calling it points and the data that I'm going to put into that array list are P Vector objects why do I want a p Vector object it's because every single frame I'm going to say points. add a new P Vector with that X Y and Z okay so I want to add a new Vector with X Y and Z to that particular array list and we can see that this is still working nothing is broken yet and what do I want to do now as those points are being added over time I want to draw all of them because I need to erase the background right so let let me just show you what I mean I'm going to move this background to draw this is actually going to produce kind of an interesting result because now you're going to see this point ah I kind of love this just moving along the pattern of the Lorenzo tractor it's kind of this beautiful interesting kind of chaotic motion I just want to stop and watch it but I'm 15 minutes in I want to see if we can wrap this up in about 20 minutes so I'm going to keep going so what I want to do is instead of just drawing the existing Point what I want to do is say for every vector v in that array list points draw a point at v.x v.y and v.z so I'm now saying instead of just drawing the current Point Loop through and this is known as an enhanced Loop in Java for every vector v in the array list points draw a point at VX v y v y v y back after a technical hiccup you can see that the entire Loren attract Vector is drawn with each element as a separate Point based on every B Vector that's in that array so really essentially I'm done there you go it's in 3D that's the entire Lorenzo tractor shape but I do want to add some more things to this and give you some exercises to think about what you might do with this so first let's just change the way we're drawing this a little bit um one thing that I could do is connect all the points with a line so instead of drawing everything as a single point I could actually set the vertex of a shape so if I say begin shape before I draw all the points and end shape after I draw all the points with vertex instead of Point what that's going to do is connect every point and you're going to see now that whoa it's doing it but it's actually trying to fill it in as if it's an entire shape so there might be some interesting things I could do with that but for now I'm just going to add no fill and we're going to be able to see now this continuous line so I could zoom into it and sort of see that every point is now connected right so now we have this continuous shape which is a nice start of sort of seeing how I could sort of Tinker with how this is visually I could also try to come up with like a clever way of changing the colors of it I'm sure that you as a wonderful person on the internet with visual design Talent will do something much more interesting but I'll just show you really quickly ah I already have that in there because I that was from the technical glitch but I could add something called color mode and I could change the color mode to HSB meaning Hue saturation brightness so what I could do is set the stroke of with a some sort of hue a saturation of 255 the maximum and a brightness of 255 so I'm now now as I draw it it's going to have a particular kind of like turquoisey greeny Hue and why not while I'm doing this um make a variable start it as Zero have that variable be the Hue and for every single vertex uh increase that by some value and then uh if it goes above 255 uh reset it back to zero and now I should get uh why not have a rainbow since this is coding rainbow uh a rainbow Loren tractor um pattern so you can see as the color is changing we've got kind of yellow red to Yellow to Orange and that sort of thing so really okay so the last thing I want to mention so what's interesting about what I'm doing with this particular strategy separate from the Lorenzo tractor um separate from the Lorenzo tractor um algorithm itself is the fact that every one of these points is saved in memory in that array so one thing I might just demonstrate to you is the fact that those points can be manipulable in real time so for example one thing I could do in processing is I could write a vector called offset uh and I could say p vector. Random 3D that's going to be give me just a random unit Vector of unit one um of length one pointed in a random Direction and I could add that offset to every single Vector in the Lorenz tractor now if I run this it's kind of going to go crazy because the unit of I've scaled up by five so you can see it's like sh shaking and jiggling like crazy so what I could do is maybe shrink that offset down quite a bit by you know make it 10% of the length of itself and you can see now if I zoom into it you can see how the points are kind of moving ever so slightly in space now I'm not suggesting that what I've done is a particularly magnificent result but what if you had all those points move along a sine wave or what if you have those points expand out and then come back to the original location or morph into another shape or fill in the blank of your interesting idea there so what I think I would like you to do now that you've seen how to program this lorenza tractor from scratch first you don't have to do anything you're just watching this video maybe you want to go outside and toss a frisbee around or eat some lunch which is what I kind of want to do right now because I'm feeling a little woozy but I think what would be interesting what I'd be excited for some of you watching this video to explore is what types of other how can you put your own design spin on this particular shape what happens if you start to manipulate the variables right here are the constants that I'm using make up your own constants look up on the Internet some other known constants for the lorenza tractor that produce other kinds of shapes what happens if you change the DT variable what can you mess around with these formulas and get something else can you go to Paul could you go to a site and find a different kind of attractor that's not the Loren one there's a sort of there's a lot of these under the sort of category of strange attractors so try to make your own version of this uh be creative play experiment don't worry too much and share with me your your results but this has been hopefully something you've enjoyed which is programming the lorenza tractor from scratch in processing in 3D thank you very much for watching I'm thrilled to have finished this because you might not know there was about a 45 minute audio problem I really hope the audio is working right now and this is actually the end of this video thanks for watching and I'll see you in the future in in another video goodbye

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