I don’t really understand the difference between an ODE and a dynamical system.
The ODE just seems like a way to describe it.
Is there more to it?
Are there dynamical systems which cannot be represented by any ODE?
[Math] Difference between an ODE and dynamical system
dynamical systemsordinary differential equationsterminology
Related Solutions
Note: this discussion is applicable to both continuous and discrete time dynamical systems, but I will limit it to continuous time as they are analogous.
Given the dynamical system $x' = f(x)$, the vector $x$ is the state of the dynamical system, and the function $f$ tells us how the system moves. In special cases, the system does not move and we call these states 'fixed or equilibrium points' of the system.
For example, lets look at:
$$x' = x^3 - 8$$
What happens when $x = 2$? The derivative $x'$ at this point is fixed, it is neither increasing nor decreasing, in other words, it is stuck on $\tilde x = 2$. If you want a fixed point at zero, just use $x' = x, x^2, ~\text{or}~ x^3 \ldots$
Do you notice any other values that would give us this case in this dynamical system? No, there is a single, unique fixed point.
Lets look at a direction field plot for this system and see if we can further describe this. The direction field plot is:
What do you notice across the entire range when $x = 2$? If you perturb just slightly away from $2$, look what happens to the solution depending on which side of $2$ you end up on.
Lets change the example to $x' = (x-1)(x-3)^2$. How many fixed points do we have in this case? Two of them, namely, $x = 1$ and $x = 3$.
A direction field plot of this case should show us fixed points are these locations, so this looks like:
Do you notice what happens to the derivative at those two fixed points?
It is also important to note that a system may have no equilibrium points, for example $x' = e^x$. So, we can have none, one (unique) or many equilibrium (fixed) points for a system.
This notion can easily be extended to higher dimensional systems too. For example, lets take the system:
$$x' = x^2 + y^2 -25 \\ y' = x + y + 1$$
In this case, to find the fixed points, we want to know where $x'$ and $y'$ are simultaneously zero. We find two points for this at: $(-4, 3)$ and $(3,-4)$. A phase portrait of this system should show us these two fixed points, so we have:
Do you notice those two points on the phase portrait?
You also asked about the pendulum. The orbits of the pendulum away from the equilibrium points $(n \pi, 0)$, where $n \in \mathbb{Z}$, are given by the solutions of the scalar equation:
$$\dfrac{dx_2}{dx_1} = \dfrac{-(g/l) \sin x_1}{x_2}$$
If we plot the contours of this, we have:
Do you see where we are getting zero? This is where the derivative is not changing.
We can write the pendulum as a system as:
$$x' = y \\ y' = -\sin x$$
A phase portrait shows:
The center corresponds to a state of neutrally stable equilibrium, with the pendulum at rest and hanging straight down. The small orbits surrounding the center represent small oscillations about the equilibrium. The critical case that correspond to heteroclinic trajectories join the saddles, which represent the inverted pendulum at rest.
This was a handwaving argument and there are formal mathematical definitions for these equilibrium points.. We can get into formal definitions, discussions about cardinality and other, but there is not enough room in the margins to include those here.
These fixed points allow us to do a lot of qualitative analyses without actually solving the system and this is extremely helpful. Once we figure out the fixed points, we want to classify them by type and this tells us about stability. We can have stable and unstable points and they are further broken down into nodes, saddles, $\ldots$.
Stability
There are several notions for stability. We say that $x* = 0$ is an attracting fixed point when we start near $x*$ and approach it as $t \rightarrow \infty$. If $x(t) \rightarrow x*$ as $t \rightarrow \infty$, we call this globally attracting.
The other notion typically used is Liapunov stability. In this approach, we say a fixed point $x*$ is Liapunov stable if all trajectories that start sufficiently close to $x*$, remain close for all time.
In practice, there are really two types of stability:
- If a fixed point is 'both' Liapunov stable and attracting, it is stable or sometimes asymptotically stable.
- $x*$ is unstable when it is neither attracting nor Liapunov stable.
It is amazing that using those two notions, we can tell all of this from the eigenvalues of a system. We look at their sign and it is able to tell us this. For example if both are positive we are unstable. Why? Because any trajectory, regardless we we start will go away from the fixed. When both are negative, we have the opposite case. When they are different signs, we have a saddle as we have this argument going on between the female and male eigenvalues and that is never good. Lastly, we can have neutral stability with a center with complex eigenvalues. These notions tell us a lot of information about the long term behavior of our system. We want to know that over time the building, bridge or whatever will be stable for all time and if we can describe a system using dynamics, we can easily make these determinations.
Again, this is a bit of hand waving from the mathematical definitions, but is about as easy an understanding as you can muster.
First let me address what your various textbooks say.
Given a manifold $M$, to say that a vector field $f$ is a mapping from $M$ to $TM$ is incomplete. Let me use $T_p M$ to denote the fiber of $TM$ over the point $p \in M$, in other words $T_p M$ is the tangent space of $M$ at $p$. A vector field on a manifold $M$ is not just any old mapping from $M$ to $TM$. Instead, a vector field on $M$ is a mapping $f : M \to TM$ such that for each $p \in M$ we have $f(p) \in T_p M$, in other words $f(p)$ is a vector in the tangent space at $p$.
In the special case where $M = \mathbb{R}^n$, if you keep all of this notation in mind, then your two textbooks are saying essentially the same thing. The tangent bundle in this case is a product $T \mathbb{R}^n = \mathbb{R}^n \times \mathbb{R}^n$, and the tangent space at each point $p\in \mathbb{R}^n$ has the form $$T_p \mathbb{R}^n = \{(p,v) \,|\, v \in \mathbb{R}^n\} $$ where the vector operations on the vector space $T_p \mathbb{R}^n$ are defined by simply ignoring the $p$ coordinate, i.e. $(p,v) + (p,w) = (p,v+w)$ and similarly for scalar multiplication. Because of this, there is a canonical isomorphism between vector fields expressed as functions $$f : \mathbb{R}^n \to \mathbb{R}^n $$ and vector fields expressed as functions $$g : \mathbb{R}^n \to T \mathbb{R}^n $$ This canonical isomorphism is given by the formula $g(p)=(p,f(p))$.
Regarding your last sentence, perhaps there may be elementary expositions of dynamical systems that restrict attention to $M=\mathbb{R}^n$, but the full theory of dynamical systems considers manifolds in all their full and general glory, and in this theory it is not sufficient to consider $\mathbb{R}^n$. Dynamical systems on spheres, on toruses, and on all kinds of manifolds are important.
I would also point out that it is misleading to say that a dynamical system on a manifold $M$ is a vector field. What is important in dynamical systems is not the vector field in particular, but its integral curves and their behavior over long time spans.
Best Answer
In general, a dynamical system is defined as a system in which a function (or a set of functions) describes the evolution of a point in a geometrical space.
The point in question may lie in a space where every coordinate is a value you want to track (for example, the current and the voltage drop at the ends of a capacitor, or the population of fish in a lake). I believe your confusion stems from the fact that dynamical systems are introduced when derivatives or integrals are introduced in systems of equations; however, the equations need not be differential in nature.
Now, ODEs are usually the simplest way to describe a dynamical system and its evolution with time. The best example here could be an RLC circuit, where everything could be described in terms of derivatives of voltages and currents. ODEs are also only one of the possible equations that describe a system. There could be Partial DEs, or even something completely different like stochastic processes, random variables, recursive definitions. ODEs are just one of the tools involved in the description of dynamical systems, and often the simplest one to solve - so simple, in fact, that sometimes you may find a closed form for the solution to the equations, but this is not always the case (and in general, a closed, analytical form does not exist for any given dynamical system).