{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": { "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [ "%matplotlib inline\n", "%config InlineBackend.figure_format='retina'" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Basics of Dynamics Systems and Bifurcation Theory\n", "# Workshop \\#1, 23 Jan. 2020\n", "\n", "#### Navid C. Constantinou, RSES, ANU, 2020" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Dynamical Systems\n", "\n", "A dynamical system can be expressed in the form\n", "\n", "$$\n", " \\frac{\\mathrm{d}}{\\mathrm{d}t} \\phi(t)= \\mathcal{F}\\big(\\phi(t),\\,t\\big) \\tag{1}.\n", "$$\n", "\n", "Above, $\\phi(t)$ includes all variables needed to describe the state of the system. (Hereafter, dots above variables denote time-differentiation.)\n", "\n", "---\n", "\n", "#### Example: the pendulum\n", "\n", "For a pendulum that evolves freely in three dimensions, $\\phi$ includes the position and the velocity of its free end. In this case, $\\phi$ has six components:\n", "$$\n", " \\phi = [x(t), y(t), z(t), \\dot{x}(t), \\dot{y}(t), \\dot{z}(t)].\n", "$$\n", "\n", "---\n", "\n", "If $\\mathcal{F}$ in Eq. (1) *does not* depend explicitly on time $t$ we say that the dynamical system is **autonomous**. Otherwise, the dynamical system is **non-autonomous**.\n", "\n", "Given equation (1) and the state of the system at some instance $t=t_0$, e.g., $\\phi(t_0)=\\phi_0$, we can predict the state of the system $\\phi(t)$ for all times $t$. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## All dynamical systems are of “first order” \n", "\n", "\n", "Any $n$-th order differential equation can be rewritten as a system of $n$ differential equations all being **first order**. In that sense, *all* dynamical systems are first-order in time (but of course their dimensionality may vary).\n", "\n", "\n", "#### Exercise 1\n", "\n", "Show that any **non-autonomous** system of $n$ differential equations corresponds to an **autonomous** system of $n+1$ differential equations.\n", "\n", "*Hint*: You can start by an example: Show that the forced 1D harmonic oscillator\n", "\n", "\\begin{align*}\n", " \\ddot{x} + 2\\gamma\\dot{x}+\\omega^2 x = f_0\\cos(2\\pi t),\n", "\\end{align*}\n", "\n", "can be rewritten as a system of 3 equations that **do not** depend explicitly on time. Then generalise this to what the exercise wants.\n", "\n", "Hint: You may need to change/introduce a new variable." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Exercise 2 (optional)\n", "\n", "Show that any **nonlinear dynamical system** can be rewritten as a **linear** dynamical system of higher dimensionality.\n", "\n", "Take as an example the system:\n", "\n", "\\begin{align*}\n", " \\dot{x} = -x^2.\n", "\\end{align*}\n", "\n", "How big is the dimension of the equivalent linear system for the example above?\n", "\n", "Food for thought: If any nonlinear system can be transformed into a linear one then what's the point of distinguish between *linear* and *nonlinear* dynamical systems?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Attractors of Dynamical Systems\n", "\n", "\n", "**Fixed points** or **equilibrium points** are special values of $\\phi=\\phi^e$ which satisfy:\n", "\n", "$$\n", "\\mathcal{F}(\\phi^e) = 0. \\tag{2}\n", "$$\n", "\n", "They are called equilibria or fixed points because if the system finds itself in one of these states then it will stay there for eternity.\n", "\n", "The character of a fixed point is studied by assuming small deviations about that fixed point,\n", "\n", "$$\n", " \\phi = \\phi^e + \\phi',\\quad\\text{where}\\quad\\|\\phi'\\|\\ll 1. \\tag{3}\n", "$$\n", "\n", "Determining how these small deviations will evolve characterises the fixed point. We do that by writing out the linear dynamical system that governs $\\phi'$. By inserting Eq. (3) in Eq. (1) and after linearization (i.e., discard of the terms that involve quantites which are quadratic or higher order in components of $\\phi'$), we get:\n", "\n", "$$\n", " \\dot{\\phi'} = \\mathcal{F}(\\phi^e + \\phi') \\approx \\mathcal{A}(\\phi^e)\\,\\phi',\\tag{4}\n", "$$\n", "\n", "where $\\mathcal{A}$ is a linear operator that depends on the fixed point $\\phi^e$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "---\n", "\n", "#### Example: the 1D pendulum\n", "\n", "