Introduction to nonlinear dynamics
1. Continuous and discrete dynamical systems
A dynamical system describes the evolution of physical or abstract state over time. In planetary dynamics, the state is the position and velocity of the Sun and the planets. Once the state is known, the future of the system is completely determined by the laws of classical mechanics and Newton's law of gravity. In a physical setting, the state is often the Euclidean space \R^n, while the evolution is given by an ordinary differential equation (ODE):
\begin{cases} \frac{dx_1}{dt} & = f_1(x_1, \cdots, x_n), \\
& \vdots \\
\frac{dx_n}{dt} & = f_n(x_1, \cdots, x_n).
\end{cases}\qquad (1.1)
An ODE is a continuous time dynamical system, as it's solution is a function of the contiuous variable t.
A discrete time dynamical system on \R^n will be given by a map
\bff \st \R^n \to \R^n,
\quad
\bff(\bx) = \bmat{f_1(x_1, \cdots, x_n) \\ \vdots \\ f_n(x_1, \cdots, x_n)}.
A discrete time system evolves by applying the map repeatedly. If the state at time k is \bx_k, then \bx_{k+1} = \bff(\bx_k). While our physical time is always continuous, discrite time systems come up in many natural settings:- We may only care about input and output, not the process in between.
- The system may be continuos-time, but we only observe it at discrete time, for example, a physical experiment where a camera takes a snapshot every second.
- A computer simulation always runs on descrete time.
Discrete time system will come up in the second half of the course; for now, our focus is on ODEs.
2. Principles of ODE
A traditional ODE course tend to focus on calculus techniques and explicit solutions. We require a different perspective in this course. Here are a few basic rules.
2.1. Most ODEs don't have explicit solutions. We should try to understand a system without solving it.
Example 2.1.
\frac{dx}{dt} = x + \sin x.
Solving the equation using the standard method requires computing the integral
\int \frac{dx}{x + \sin x}
which seems difficult. (I don't know how to compute it). However, if we look at the direction field of the equation vs that of \frac{dx}{dt} = x, we see they are very similar.
Figure 1.
To understand the system, the precise formula of the solution is not as useful as a global picture that describes all the solutions in a region. We should seek a geometrical and qualitative description of the solution.
2.2. Chain rule applied to an abstract function is the most important skill in this course
Since differential equation is an equation about the derivative of an (abstract, unknown) function, one can compute with the derivatives even if we don't know what the functin is. Maybe 80 percent of the tricks we use involve this.
For example, to solve the equation
\frac{dx}{dt} = y, \quad \frac{dy}{dt} = -x,
we consider any solution x(t), y(t) of the system, and compute the derivative of the expression
x^2(t) + y^2(t).
It turns out that
\frac{d}{dt} \left( x^2(t) + y^2(t)\right)
= 2 x \frac{dx}{dt} + 2y \frac{dy}{dt}
= \text{ (using the equation!) }
2x (y) + 2y(-x)
= 0.
Therefore the x^2(t) + y^2(t) where x(t), y(t) solves the equation must be a constant function of t. Therefore any solution of the equation must lie on a circle x^2 + y^2 = C.2.3. All ODEs are first order, and therefore can be understood from a phase portrait
This is a simple trick, but it allows us to completely shift our perspective. Take, for example, this second order equation in dimension one:
\frac{d^2 x}{dt^2} + x = 0.
By setting
y = \frac{dx}{dt},
we get
\frac{dx}{dt} = y, \quad \frac{dy}{dt} = \frac{d^2 x}{dt^2} = - x.
We traded a second order, one-dimensional system for a first order, two-dimensional system. This can be done for arbitrary number of derivatives. This means any ODE can be rewritten in the form (1.1). In vector form, we have
\frac{d}{dt} \bx = \bfF(\bx),
where \bfF: \R^n \to \R^n is a vector field. The solutions are given by curves (in \R^n) tangent to the vector field.2.4. Often, we want more than understanding one system. We want to understand a range of systems as we vary parameters.
In applications, the equation is often prescribed by laws of nature, while the parameters corresponds to measurable quantities. The nature of the systems can differ significantly over these quantities. A contemporary example is the infection disease model: the infection rate is a crucial parameter that can dictate the outcome of a system.
The exact value of the parameter, crossing of which causing the nature of the system to change, is called a bifurcation value. The study of bifurcations is one of the major themes of applied dynamical systems.
3. Examples
Notation: we will start using the "dot" notation for derivative, namely
\dot{x} = \frac{dx}{dt}, \quad \ddot{x} = \frac{d^2 x}{dt^2}, \text{ etc.}
Example 3.1. (The Harmonic Oscilator).
The model for a free moving object attached to a spring. Acoording to Newton's second law and Hooke's law, we have
\ddot{x} = - a x,
where a > 0 is a parameter that measures the strength of the spring.
Figure 2.
Set y = \dot{x}, we get
\dot{x} = y, \quad \dot{y} = - ax.
We observe that \frac{d}{dt}\left( y^2 + a x^2\right) = 0, hence the phase portrait consists of ellipses y^2 + a x^2 = C.
Figure 3.
Example 3.2. (The mathematical pendulum).
Use x to denote the angle between the pendulum and the vertical direction, the equation of the pendlulum can be written as
m l \ddot{x} = - mg \sin x, \quad
\text{ or }
\quad
\ddot{x} = - \frac{g}{l} \sin x.
As a system:
\dot{x} = y, \quad \dot{y} = - \frac{g}{l} \sin (x).
Figure 4.
We will be analyzing this system later; for now, let's take a look at the phase portrait. Can you interpret how the different trajectory lines correspond to in the physical system?
Figure 5.
Question 3.3.
An equilibrium is a trajectory of the system that stays constant over the time. For the pendulum, this means that \dot{x} = \dot{y} = 0. In the original physical system, this means both the position and the velocity stay constant.
Take a lot at the pendulum phase portrait, can you identify the equilibria?
A stable equilibrium is one that a small change to the initial condition leads to an orbit that is close to the orginial equilibrium. An unstable equilibrium is one that a small perturbation causes the trajectory to go far away.
Classify the pendulum equilibria as stable or unstable.
Remark 3.4.
We will later define stability rigorously, and prove theorems about stability.
Example 3.5. (The double pendulum).
Consider two penduli joined together. The system can be described by two angles x_1 and x_2. The equation is actually quite complicated, but it can be given by a system of two second order equations:
\ddot{x}_1 = f_1(x_1, x_2), \quad
\ddot{x}_2 = f_2(x_1, x_2).
After setting y_1 = \dot{x}_1, y_2 = \dot{x}_2, this is a 4-dimensional system.
Figure 6.
On large amplitude, the double pendulum can exhibit chaotic behaviour. See link for a simulation.