{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Numerical chaos\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/numerical-chaos). This notebook includes the lesson text and its Python examples in reading order. Run the setup cell, then run the remaining cells from top to bottom. In Colab, use File → Save a copy in Drive to keep your edits. You can run and edit this view without changing the course source. Examples use illustrative data unless stated otherwise." ] }, { "cell_type": "code", "id": "code-001", "metadata": {}, "source": [ "import importlib.metadata\n", "import subprocess\n", "import sys\n", "\n", "requirements = {\"numpy\": (2, 0), \"sympy\": (1, 13), \"matplotlib\": (3, 8)}\n", "to_install = []\n", "for package, minimum in requirements.items():\n", " try:\n", " installed = importlib.metadata.version(package)\n", " version = tuple(int(part) for part in installed.split(\".\")[:2])\n", " except importlib.metadata.PackageNotFoundError:\n", " version = (0, 0)\n", " if version < minimum:\n", " to_install.append(f\"{package}>={minimum[0]}.{minimum[1]}\")\n", "if to_install:\n", " subprocess.check_call([sys.executable, \"-m\", \"pip\", \"install\", *to_install])\n", "\n", "import numpy as np\n", "import sympy as sp\n", "import matplotlib\n", "import matplotlib.pyplot as plt\n", "\n", "print(\"Python\", sys.version.split()[0])\n", "print(\"NumPy\", np.__version__)\n", "print(\"SymPy\", sp.__version__)\n", "print(\"Matplotlib\", matplotlib.__version__)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-002", "metadata": {}, "source": [ "Chaos is deterministic behavior with sensitive dependence on initial conditions and\n", "structured long-term dynamics. Numerical calculations can reveal it, but unstable time\n", "steps, roundoff, inconsistent sampling, or hidden randomness can also make trajectories\n", "diverge. A chaos claim therefore needs controls, not just an unpredictable-looking curve.\n", "\n", "## A nonlinear driven example\n", "\n", "An ideal harmonic oscillator is linear and does not exhibit deterministic chaos. A\n", "periodically driven, damped nonlinear pendulum can. One common dimensionless form is\n", "\n", "\n", "$$\n", "\\ddot\\theta+q\\dot\\theta+\\sin\\theta=f\\cos(\\Omega t)\n", "$$\n", "\n", "\n", "Here $q$ controls damping,\n", "$f$ controls forcing,\n", "and $\\Omega$\n", "sets drive frequency. The sine term is nonlinear. Replacing it with\n", "$\\theta$ gives a linear driven\n", "oscillator and removes the nonlinear route to chaos.\n", "\n", "The first-order state includes angle, angular velocity, and drive phase:\n", "\n", "\n", "$$\n", "\\dot\\theta=\\omega\\qquad\\dot\\omega=-q\\omega-\\sin\\theta+f\\cos\\phi\\qquad\\dot\\phi=\\Omega\n", "$$" ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "\n", "def derivative(state):\n", " angle, velocity, phase = state\n", " return np.array(\n", " [velocity, -0.5 * velocity - np.sin(angle) + 1.2 * np.cos(phase), 2 / 3]\n", " )\n", "\n", "\n", "def rk4(state, dt):\n", " a = derivative(state)\n", " b = derivative(state + dt * a / 2)\n", " c = derivative(state + dt * b / 2)\n", " d = derivative(state + dt * c)\n", " return state + dt * (a + 2 * b + 2 * c + d) / 6\n", "\n", "\n", "print(derivative(np.array([0.2, 0.0, 0.0])))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "Including phase makes it clear that two identical angle-velocity pairs at different drive\n", "phases are not the same complete state.\n", "\n", "## Determinism and sensitivity\n", "\n", "Deterministic means the same complete state and parameters produce the same derivative.\n", "Sensitive dependence means nearby initial states can become macroscopically separated.\n", "During an early interval, separation may behave approximately as\n", "\n", "\n", "$$\n", "d(t)\\approx d(0)e^{\\lambda t}\n", "$$\n", "\n", "\n", "with positive Lyapunov rate\n", "$\\lambda$. The trajectories obey\n", "the same equations; uncertainty in the initial digits grows.\n", "\n", "This is not randomness. With identical floating-point input and execution policy, a\n", "deterministic program should replay. Nor does every nonlinear system become chaotic:\n", "many parameter ranges have fixed, periodic, or quasiperiodic attractors.\n", "\n", "## Construct a controlled pair\n", "\n", "Copy the complete reference initial state and perturb one component. If angle changes by\n", "$\\epsilon$, verify that\n", "the initial distance equals that value under the chosen norm." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "reference = np.array([0.2, 0.0, 0.0])\n", "perturbed = reference.copy()\n", "perturbed[0] += 1e-7\n", "assert perturbed[2] == reference[2]\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "If random forcing is not part of the model, do not add it. If it is included, paired runs\n", "need the same saved random sequence unless stochastic variation is the intended\n", "experiment.\n", "\n", "## Define a meaningful distance\n", "\n", "Angle and angular velocity have different units. A normalized phase-space distance can\n", "be\n", "\n", "\n", "$$\n", "d(t)=\\sqrt{\\left(\\frac{\\Delta\\theta}{\\Theta_0}\\right)^2+\\left(\\frac{\\Delta\\omega}{\\Omega_0}\\right)^2}\n", "$$\n", "\n", "\n", "where $\\Theta_0$ and\n", "$\\Omega_0$ are\n", "declared characteristic scales. For angular coordinates, use the shortest wrapped\n", "difference when angles differing by\n", "$2\\pi$ represent the same\n", "orientation.\n", "\n", "Different norms can change finite numerical rates, though a well-established largest\n", "Lyapunov exponent is less sensitive to reasonable smooth norm choices. Report the norm\n", "and scales.\n", "\n", "## Read logarithmic separation carefully\n", "\n", "Taking a logarithm gives\n", "\n", "\n", "$$\n", "\\ln\\frac{d(t)}{d(0)}\\approx\\lambda t\n", "$$\n", "\n", "\n", "Fit only where distance is above roundoff and solver error, remains small enough for local\n", "growth, and lies below attractor-size saturation. Once trajectories explore unrelated\n", "parts of a bounded attractor, distance stops growing exponentially and fluctuates around\n", "a system scale. The plateau is not part of the early slope.\n", "\n", "A finite-window estimate is\n", "\n", "\n", "$$\n", "\\lambda_T=\\frac{1}{T}\\ln\\frac{d(T)}{d(0)}\n", "$$\n", "\n", "\n", "but one endpoint ratio can be misleading when distance oscillates. Fit a justified window\n", "or use periodic renormalization and report variation across windows and initial states.\n", "\n", "## Renormalized Lyapunov estimation\n", "\n", "To avoid saturation, evolve a nearby state for a short interval, measure its separation,\n", "record logarithmic growth, and rescale the difference back to a small initial length in\n", "the same direction." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "initial_size, log_growth, dt = 1e-7, 0.0, 0.01\n", "for interval in range(100):\n", " for _ in range(100):\n", " reference, perturbed = rk4(reference, dt), rk4(perturbed, dt)\n", " difference = perturbed - reference\n", " difference[0] = (difference[0] + np.pi) % (2 * np.pi) - np.pi\n", " size = np.linalg.norm(difference[:2])\n", " if size <= 0:\n", " raise RuntimeError(\"Perturbation was lost to rounding.\")\n", " log_growth += np.log(size / initial_size)\n", " perturbed = reference + difference * initial_size / size\n", "print(\"Finite-time growth estimate:\", log_growth / 100)\n", "print(\"Repeat with smaller steps and longer records before interpreting it.\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "Angle wrapping, scale normalization, transient removal, interval length, and total\n", "duration all affect the finite estimate.\n", "\n", "## Poincare sections\n", "\n", "Continuous phase portraits of driven systems include the drive cycle. Sample once per\n", "drive period at the same phase:\n", "\n", "\n", "$$\n", "t_n=t_0+n\\frac{2\\pi}{\\Omega}\n", "$$\n", "\n", "\n", "A settled period-one response produces one repeated point. Period two produces two.\n", "Repeated period doublings can precede chaos, where the section contains a complicated\n", "bounded set. Quasiperiodic motion can produce a smooth closed curve rather than a finite\n", "set." ] }, { "cell_type": "code", "id": "code-009", "metadata": {}, "source": [ "period = 2 * np.pi / (2 / 3)\n", "state = np.array([0.2, 0.0, 0.0])\n", "points = []\n", "for cycle in range(20):\n", " for _ in range(1000):\n", " state = rk4(state, period / 1000)\n", " points.append([(state[0] + np.pi) % (2 * np.pi) - np.pi, state[1]])\n", "print(\"First Poincare points:\", points[:3])\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "Use interpolation or event location so every point has the same drive phase. Timing error\n", "can turn one periodic point into an artificial cloud.\n", "\n", "## Parameter sweeps and bifurcations\n", "\n", "A bifurcation diagram varies one parameter, removes transients, and records a consistent\n", "observable such as Poincare angular velocity. Each parameter value needs enough time to\n", "settle or reveal long behavior.\n", "\n", "Restarting each value from a common initial state and continuing from the previous final\n", "state are different experiments. Nonlinear systems can have coexisting attractors, so\n", "sweep direction and initial condition can change the displayed branch. Preserve both in\n", "the result.\n", "\n", "A finite sample can make a long-period orbit look chaotic or a chaotic attractor look\n", "periodic. Increase retained cycles and compare structure under parameter-grid refinement.\n", "\n", "## Physical chaos versus numerical instability\n", "\n", "An unstable integrator can also create rapid separation. To distinguish it, repeat at\n", "smaller time steps or tighter tolerances using common physical output times. A physical\n", "early-growth rate and Poincare structure should approach a consistent result over a\n", "justified time interval.\n", "\n", "Use controls:\n", "\n", "- Zero perturbation should replay exactly under deterministic execution.\n", "- A linear damped oscillator should not show a false positive Lyapunov rate.\n", "- The nonlinear system in a known periodic parameter range should yield a finite\n", " Poincare cycle.\n", "- An intentionally excessive step should fail or visibly change under refinement rather\n", " than being accepted as chaos.\n", "\n", "For the driven damped pendulum, mechanical energy is not conserved because drive adds\n", "energy and damping removes it. Still track their power ledger. Unexplained energy change\n", "beyond those transfers indicates numerical or implementation error.\n", "\n", "## Floating-point and update-order effects\n", "\n", "Roundoff supplies tiny perturbations. Chaotic dynamics can amplify them, so bitwise\n", "agreement across hardware or operation ordering may end after some time. This does not\n", "make all outputs useless. Statistical distributions, attractor geometry, Lyapunov rates,\n", "and bifurcation locations may remain reproducible within uncertainty.\n", "\n", "Do not update one component or coupled oscillator from a newer time while another still\n", "uses the old state unless the integrator explicitly defines that sequence. Hidden update-\n", "order dependence seeds a different numerical method.\n", "\n", "## Predictability and uncertainty\n", "\n", "If initial uncertainty is\n", "$d_0$ and the maximum useful\n", "uncertainty is $d_{\\max}$,\n", "an approximate predictability horizon is\n", "\n", "\n", "$$\n", "t_{\\mathrm p}\\approx\\frac{1}{\\lambda}\\ln\\frac{d_{\\max}}{d_0}\n", "$$\n", "\n", "\n", "Improving initial precision extends the horizon logarithmically, not indefinitely. Model\n", "parameter uncertainty and measurement noise must also be propagated.\n", "\n", "## Evidence for a chaos claim\n", "\n", "Publish the equation, parameters, dimensional scaling, initial state, perturbation,\n", "distance norm, precision, solver, step policy, transient duration, sampling phase, fit\n", "window, saturation threshold, and refinement results. Retain raw time series rather than\n", "only a final phase picture.\n", "\n", "One positive-looking slope or irregular trace is suggestive, not decisive. Agreement\n", "among paired separation, Poincare structure, parameter behavior, control systems, and\n", "numerical refinement makes the claim much stronger and defines exactly what is\n", "predictable over which interval.\n", "\n", "## A worked Python notebook\n", "\n", "Take a slice of the driven pendulum at zero angular velocity and drive phase zero. The question for this calculation is precise: how does **instantaneous angular acceleration**\n", "change when we change **angle**, while keeping the other choices fixed?\n", "The earlier sections explain the physical relationship. Here we turn one part of that\n", "relationship into a calculation that you can run, inspect, and change in Google Colab.\n", "\n", "The full dynamics also evolves angular velocity and drive phase; this slice alone cannot demonstrate chaos. Keep that sentence beside the results. A formula can return a number outside\n", "its useful range, and a plotting library will usually draw that number without asking\n", "whether the physical assumptions still hold. The selected interval is from\n", "-3.14159 to 3.14159 rad. It gives us a specific domain to discuss\n", "instead of leaving the graph's range to a software default.\n", "\n", "Use the notebook link above the lesson, run the setup cell, and then run the cells in\n", "order. The code below is also included there. NumPy handles the arrays of numerical\n", "values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these\n", "libraries or notebook cells are unfamiliar, start with\n", "[Your first physics notebook](https://tirta.co/a-byte-of-physics/your-first-physics-notebook).\n", "\n", "### State the relationship before choosing a graph\n", "\n", "In this small example, `q` stands for angle, expressed in rad. The output is\n", "instantaneous angular acceleration, expressed in scaled angular units. The equation below uses the numerical values in those\n", "units. Any fixed coefficients therefore carry the units needed to make the relation\n", "consistent. A value supplied in another unit must be converted before it enters this\n", "expression.\n", "\n", "\n", "$$\n", "f(q) = 1.2 - \\sin{\\left(q \\right)}\n", "$$\n", "\n", "\n", "This is a function of one changing input. It is not a statement that the physical\n", "system has only one variable. We deliberately hold the other inputs fixed so that one\n", "cause of variation is visible. Changing several inputs at once would make it harder\n", "to tell which change produced the output difference. Write down the held-fixed choices\n", "before trying a different experiment.\n", "\n", "At 0.2 rad, the expression gives 1.00133 scaled angular units. This is our\n", "reference calculation. It is a prediction of the stated example, not a new measurement.\n", "Its displayed digits help us compare implementations. They should not be read as a\n", "claim that physical inputs are known to six significant figures." ] }, { "cell_type": "code", "id": "code-011", "metadata": {}, "source": [ "import numpy as np\n", "import sympy as sp\n", "import matplotlib.pyplot as plt\n", "\n", "q = sp.Symbol(\"q\", real=True)\n", "expression = sp.sympify(\"-sin(q)+1.2\", locals={\"q\": q})\n", "reference_input = 0.2\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"rad\")\n", "print(\n", " \"Predicted instantaneous angular acceleration:\",\n", " reference_output,\n", " \"scaled angular units\",\n", ")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "metadata": {}, "source": [ "`sp.Symbol` declares a mathematical symbol. It does not assign a numerical measurement\n", "to that symbol. `subs` then replaces the symbol with our chosen input, and `float`\n", "converts the resulting symbolic number into a standard Python numerical value.\n", "Keeping these steps separate lets us inspect the equation before numerical substitution.\n", "It also prevents a later plot from becoming the only surviving record of the model.\n", "\n", "### Compare a few cases you can explain\n", "\n", "Before drawing hundreds of points, calculate three. The lower endpoint gives\n", "1.2 scaled angular units. The midpoint, at 0 rad, gives\n", "1.2 scaled angular units. The upper endpoint gives 1.2 scaled angular units.\n", "Read these together with the assumptions: they describe different choices of angle\n", "within the same example, with the other inputs held fixed.\n", "\n", "These three values are a useful first check, but they do not establish the shape between\n", "them. Two endpoints can be equal even when the curve bends or oscillates in between.\n", "Likewise, a large change between endpoints does not tell us whether the relationship is\n", "linear. We need the equation and intermediate samples to answer those questions." ] }, { "cell_type": "code", "id": "code-013", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([-3.141592653589793, 0.0, 3.141592653589793])\n", "outputs = np.asarray(evaluate(inputs), dtype=float)\n", "outputs = np.broadcast_to(outputs, inputs.shape)\n", "for input_value, output_value in zip(inputs, outputs):\n", " print(f\"{input_value:.6g} rad -> {output_value:.6g} scaled angular units\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "metadata": {}, "source": [ "`lambdify` turns the symbolic expression into a numerical function. We explicitly select\n", "NumPy as its numerical backend so it can accept an array. The same expression now has\n", "two useful forms: one for algebra and one for calculating many cases. The assertion\n", "checks that the chosen inputs produce finite outputs. It would catch an infinity or an\n", "undefined numerical result, but a finite value can still be physically inappropriate.\n", "\n", "### Make a graph that answers the question\n", "\n", "The horizontal axis shows angle; the vertical axis shows instantaneous angular acceleration. Every plotted\n", "point is a prediction of the same expression. We mark the reference case separately\n", "so that a reader can connect the earlier printed value to its place on the curve.\n", "The axis labels include units because a shape without a scale cannot tell us the size\n", "of a physical effect." ] }, { "cell_type": "code", "id": "code-015", "metadata": {}, "source": [ "input_grid = np.linspace(-3.141592653589793, 3.141592653589793, 301)\n", "output_grid = np.asarray(evaluate(input_grid), dtype=float)\n", "output_grid = np.broadcast_to(output_grid, input_grid.shape)\n", "fig, ax = plt.subplots(figsize=(7, 4))\n", "ax.plot(input_grid, output_grid, label=\"Stated model\")\n", "ax.scatter([reference_input], [reference_output], label=\"Reference case\", zorder=3)\n", "ax.set_xlabel(\"Angle (rad)\")\n", "ax.set_ylabel(\"Instantaneous angular acceleration (scaled angular units)\")\n", "ax.legend()\n", "ax.grid(alpha=0.25)\n", "fig.tight_layout()\n", "plt.show()\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "Look for a sign change, a turning point, or a region where the curve becomes steep.\n", "Connect each feature to the expression rather than to the colour of the line. If the\n", "curve has a minimum or maximum, changing the input in opposite directions may have\n", "similar effects on the output near that point. If it is nearly straight over a short\n", "interval, a local linear approximation may be useful even when the complete expression\n", "is nonlinear.\n", "\n", "Increasing the number of samples makes the line smoother and may reveal features that\n", "a coarse grid missed. It does not improve the physical inputs or enlarge the model's\n", "validity range. For quantities that can take only discrete values, use the line as a\n", "guide to the relationship and interpret only the allowed input values. A plotting\n", "function does not enforce such restrictions for you.\n", "\n", "### Measure the local sensitivity\n", "\n", "A derivative asks a more specific question than the whole graph: how quickly does\n", "instantaneous angular acceleration change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 0.2 rad is\n", "-0.980067 in those ratio units. Locally, the output decreases as the\n", "input increases. This statement is local; the sign or size of the slope can be different\n", "elsewhere on the curve.\n", "\n", "\n", "$$\n", "f'(q) = - \\cos{\\left(q \\right)}\n", "$$\n", "\n", "\n", "For a small input change, multiply the derivative by that change to estimate the output\n", "change. Then compare that estimate with a fresh evaluation of the full expression.\n", "The distinction between these two calculations matters: the derivative gives a tangent\n", "approximation, while substitution evaluates the chosen expression at the new point." ] }, { "cell_type": "code", "id": "code-017", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 0.23141592653589793\n", "linear_prediction = reference_output + slope_at_reference * (\n", " changed_input - reference_input\n", ")\n", "full_prediction = float(evaluate(changed_input))\n", "print(\"Symbolic derivative:\", slope_expression)\n", "print(\"Tangent prediction:\", linear_prediction, \"scaled angular units\")\n", "print(\"Full expression:\", full_prediction, \"scaled angular units\")\n", "print(\n", " \"Approximation difference:\",\n", " full_prediction - linear_prediction,\n", " \"scaled angular units\",\n", ")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-018", "metadata": {}, "source": [ "Here the changed input is 0.231416 rad. The tangent estimate is\n", "0.970541 scaled angular units, while the full expression gives\n", "0.970644 scaled angular units. If the two values are close, it means the chosen\n", "input change is small enough for this local approximation at this point. It does not\n", "mean the entire model is linear. If the slope is zero, a quadratic or higher-order term\n", "may supply the leading change instead.\n", "\n", "This is also the beginning of uncertainty propagation. If the input has a small stated\n", "standard uncertainty, the magnitude of the slope times that uncertainty estimates one\n", "contribution to output uncertainty. Do not automatically use the same rule for a wide\n", "interval, a discontinuity, a turning point with zero first derivative, or several\n", "correlated uncertain inputs. Those situations need more of the model than one slope.\n", "\n", "### Check the derivative with numerical differences\n", "\n", "We can estimate the same slope from values on either side of the reference point.\n", "Subtract the lower output from the upper output and divide by twice the input step.\n", "Parentheses are essential: the entire output difference must be divided by the entire\n", "input interval. Without them, Python can execute a different mathematical expression\n", "while still producing a plausible number.\n", "\n", "The calculation below uses three decreasing step sizes. Each estimate is compared with\n", "the symbolic derivative, so the reference is independent of the numerical difference\n", "formula. Near an endpoint, it is acceptable here to evaluate a small extension only\n", "because the displayed mathematical expression is smooth there; such a check does not\n", "assert that extrapolated values describe additional physical cases." ] }, { "cell_type": "code", "id": "code-019", "metadata": {}, "source": [ "input_span = 6.283185307179586\n", "for fraction in [1e-2, 1e-3, 1e-4]:\n", " step = input_span * fraction\n", " upper = float(evaluate(reference_input + step))\n", " lower = float(evaluate(reference_input - step))\n", " numerical_slope = (upper - lower) / (2 * step)\n", " difference = numerical_slope - slope_at_reference\n", " print(f\"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-020", "metadata": {}, "source": [ "For a smooth function, reducing the step often reduces the approximation error at first.\n", "Very small steps can eventually make subtraction of nearly equal floating-point values\n", "less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare\n", "the observed behaviour over a useful range of steps instead of choosing the smallest\n", "number the computer can represent.\n", "\n", "An absolute slope difference is useful when the expected slope is near zero. A relative\n", "difference would divide by a very small reference and could look enormous even when the\n", "absolute discrepancy is tiny. Away from zero, a relative difference can help compare\n", "quantities with very different scales. The tolerance should reflect the question and\n", "the numerical method, rather than a universal decimal threshold copied into every test.\n", "\n", "### Change the example and preserve the reasoning\n", "\n", "Make one copy of the reference calculation before editing it. Choose a second value of\n", "angle inside the stated interval, predict whether instantaneous angular acceleration will rise or fall,\n", "and then evaluate the expression. Place the prediction and the result in a text cell.\n", "If they disagree, identify whether the mistake was in the sign, the unit conversion,\n", "the algebra, or the assumption about how the curve behaves.\n", "\n", "Next choose a shorter plotting interval around the reference point. Compare the local\n", "curve with its tangent approximation. Then widen the interval again. The difference\n", "shows why a useful local estimate need not describe the whole range. Keep the axis\n", "labels and numerical scale visible when comparing the plots; automatic axis limits can\n", "make two different effects appear equally large.\n", "\n", "Finally, return to the physical limit stated at the start: The full dynamics also evolves angular velocity and drive phase; this slice alone cannot demonstrate chaos. Describe one change\n", "to the setup that would violate that limit. State what new input, equation, or observation\n", "would be needed before trusting an answer in the changed situation. This last step\n", "connects the Python experiment to the physics instead of treating successful execution\n", "as the only criterion for a useful result.\n", "\n", "Before saving, restart the runtime and run all cells in order. A result that depends on\n", "an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep\n", "unrounded values for later calculations, round only the reported output, and label any\n", "new data as measured, assumed, or generated. The graph and printed values should remain\n", "traceable to the same expression and the same choices of inputs.\n", "\n", "### Recover the output change by integrating the slope\n", "\n", "There is another way to check that the expression and its derivative describe the same\n", "relationship. Add up the small output changes predicted by the slope across the chosen\n", "input interval. In the continuous limit, this integral equals the output at the upper\n", "endpoint minus the output at the lower endpoint. The result has units of scaled angular units:\n", "the input unit in the integration cancels the input unit in the denominator of the slope.\n", "\n", "This statement is about the mathematics of this particular relationship. It does not\n", "assign a new physical meaning to an arbitrary area under a graph. For example, an area\n", "under force versus distance represents work, while an area under a different pair of\n", "quantities may represent something else or have no standard name. Always carry the\n", "axis units through the calculation before naming an integral." ] }, { "cell_type": "code", "id": "code-021", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(3.141592653589793) - evaluate(-3.141592653589793))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(-3.141592653589793, 3.141592653589793, points)\n", " slopes = np.asarray(slope_function(grid), dtype=float)\n", " slopes = np.broadcast_to(slopes, grid.shape)\n", " integrated_change = np.trapezoid(slopes, grid)\n", " print(\n", " points,\n", " \"integrated change:\",\n", " integrated_change,\n", " \"endpoint change:\",\n", " endpoint_change,\n", " )\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-022", "metadata": {}, "source": [ "The trapezoidal rule joins neighbouring slope samples with straight segments and adds\n", "their signed areas. A negative contribution remains negative; replacing every slope\n", "by its magnitude would calculate a different quantity. Increasing the sample count\n", "provides a practical refinement check. If the values do not settle toward the endpoint\n", "difference, first inspect the interval, singularities, and array ordering.\n", "\n", "Do not conclude that two agreeing calculations prove the model is physically complete.\n", "Both use the same selected relationship, so an omitted effect would be omitted from\n", "both. Their agreement is evidence that the numerical tools implement that relationship\n", "consistently. Comparing with an independent experiment or a more appropriate physical\n", "model is a separate test.\n", "\n", "Keep this comparison in the notebook when you change the expression. It is particularly\n", "useful after adding a coefficient, changing a sign convention, or converting an input\n", "unit. A code cell that produces a smooth graph can conceal these mistakes; a second\n", "calculation of the same output change gives you another place to notice them.\n", "\n", "## Sources\n", "\n", "- [OpenStax, *University Physics Volume 1*, Sections 15.4-15.6, pendulums, damping, and forcing](https://openstax.org/books/university-physics-volume-1/pages/15-4-pendulums).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 23, \"Simple Harmonic Motion\"](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter23/).\n", "- [Benjamin Crowell, *Light and Matter*, nonlinear dynamics and pendulum discussions](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [Physics LibreTexts, \"The Road to Chaos\"](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Graduate_Classical_Mechanics_%28Fowler%29/23%3A_Damped_Driven_Pendulum-_Period_Doubling_and_Chaos/23.02%3A_The_Road_to_Chaos).\n", "- [Physics LibreTexts, \"Damped Driven Pendulum\"](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Graduate_Classical_Mechanics_%28Fowler%29/23%3A_Damped_Driven_Pendulum-_Period_Doubling_and_Chaos/23.01%3A_Damped_Driven_Pendulum)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "numerical-chaos.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/numerical-chaos", "source_sha256": "04a3186714f9b2ca368d338d4bb665212283358c5bccb568f9b1c864eb4f6bc4" } }, "nbformat": 4, "nbformat_minor": 5 }