I am not sure what you mean specifically by "prediction science" but if you mean deterministic, then the answer is deterministic systems are a subset of dynamical systems. Any system which evolves are known as dynamical in nature. Now, if using initial conditions we can predict/determine its future state, we call such system deterministic. Examples of non-deterministic dynamical systems can be chaotic system, whose evolution cannot be predicted using initial conditions.

Modeling is part of prediction science. Dynamical system investigate about futur results for any phenomena according to known data. I think this is what you mean by prediction science.

I do not quite agree with the above statements, even though they are correct. Certainly, a non-dynamic system would be static and therefore quite trivial to "predict". On the other hand, even some amazingly simple-looking systems have a very unpredicatble behaviour (for example, try and google "double pendulum").

But the point where I deviate from the previous answers has more to do with focus: we often study dynamic systems simply because we want to know how they work, and the reason need not be any prediction. We want to model their Hamiltonian, be it classical or quantum, understand its terms, etc. Prediction (in time or space, forward or backward) is something that might come later - or it might never really get into focus. Take a vortex in a liquid bath: its dynamics is fascinating to study for its own sake, even though in most situations its future is short-lived and rather predictable. Or take a forensic expertise centering on whether a certain event could or could not happen.

In any case, regardless of the exact meaning of the term "prediction", we need to know the dynamics of a system quite well before making any predictions about it. It follows that the "science of dynamic systems" precedes and ENABLES the "science of predicting". Predicting is only one of the many goals of Science and, in my opinion, not the most important one.

Although not completely sure, I think that the first systematization of the concept of prediction was done by N. Winer in his book "Extrapolation, interpolation, and smoothing of stationary time series, with engineering applications", yellow cover book that became known as Yellow Hazard, because it was very difficult to read and understand. However, he could implement a predictor of trajectory of fighters in the 2nd great war. Only after the war, his collaborator N. Levinson presented a discrete-time version that made clear the relation of the linear predictor and the autoregressive (AR) modeling. In fact, the coefficients of the predictor are the negative of the AR coefficients. In the middle, Levinson proposed the famous Levinson recursion for Toeplitz matrices.

Consider the state-space representation of a system. The state equation or dynamic equation is a prediction equation. This is easily seen in the discrete-time formulation:

x(n+1) = f(A,x,n) + Bu(n)

Another place where we find prediction is in the feedback systems. The feedback loop makes a prediction of the reference and the prediction error is the input to the plant (open loop).

Just to add my two-cents worth. It is not clear to me to what extent, or degree, the details of a dynamical system are needed to be know in order to make important predictions. It seems to me that that is what Topology is all about. For example, consider René Thom's Catastrophe theory (see for example: https://www.math.hmc.edu/~thompson/F13/zeeman.pdf), and I don't pretend to be up on this, but it seems that for certain processes (controlled by no more than 4 factors) there are 7 elementary catastrophes.

I guess I'm a bit confused about this question. A dynamic system is one where the state of the system evolves over time in a determinist manner. It can be described of a ordinary differential equation, a flow or then the mapping or the first recurrent map (a.k.a. Poinacre map). Formally in finite dimensions a dynamical systems is a continuous mapping from RxX ->X where X is an appropriate topological space where f(t,0)=x and f(t,f(s,x))=f(s+t,x), the group property. If X is equipped with other structure, like a differential structure, the f will normally be required to be differential. Of course R can be replaced with the integers to get dynamical systems of maps.

In these cases the future state of the system on finite or infinite dimensions are know from the initial condition and the state transition. There is a rich theory for behavior of such systems, attractors, stable and unstable manifolds of hyperbolic points, etc. While someone mentioned chaotic behavior - it's not that such behavior is random it is that it the evolution is not stable with respect to the initial conditions. That a system is chaotic at x0 if there is no neighborhood U of x0 such that that if x is in U then the system that starts at x0 and x will convert. Another description is there are strange attractors - attractors that have a fractal structure.

Such flows will occur ordinary differential equations but can arise as solutions of partial differential equations. The Smale horseshoe and then Smale's description of symbolic dynamics is the prototypical description of a chaotic dynamical system.

Ergodic dynamical systems are systems exhibiting "randomness" and also exhibits flows, e.g. the Bernoulli flow. The Fokker-Planck equation a.k.a. Kolmogorov forward equation is that describes the probability of the time evolution of a Brownian motion. In such systems the evolution is describes the probability density function of the state as it evolves over time.

The issue someone raised about Thom's description of

Catastrophes is more akin the the stabilities of equilibrium points (critical points of a mapping) and is one feature of non-linear dynamics (non-linear differential equations with the double pendulum being a good example). Catastrophe theory comes out of singularity theory which is the study of spaces that are manifolds except for a singular point, e.g., instead of being smooth - there is a point singularity or cusp or fold, etc. https://en.wikipedia.org/wiki/Singularity_theory

The topological prosperities at a singular point can be very different than at a non-singular point. Singular point arise in a lot of places - in the study of algebraic curves for example, (0,0) is a double point for the algebraic curve defined by y^2=x^2 + x^3 and is a cusp of y^2=x^3. The study of critical points also gives rise to singular points.

It is also closely related to what is called a bifurcation. That is if one has a ordinary differential equation x'=f(x,mu), where ' is the t-derviative and mu as a parameter over some open. Call the solution x(t,mu). It could very well happen that there is a point mu0 such that the behavior of x(t,mu) has the same behavior as t->infinity mu in a neighborhood of mu0. It could also happen that x(t,mu) has completely different behavior for mum0. Such a mu0 is a bifurcation of the system.

https://en.wikipedia.org/wiki/Bifurcation_theory

Dynamical systems give rise to a rich theory in mathematics about the structure of how systems that evolve over time behave and been along investigated. My confusion comes from what is meant by "prediction science."

Truman, I have the same problem as you.

I feel that something like "prediction science" might exist. Intuitively, I can see that it makes sense to study various approaches to predictions, their types, expected errors, etc. But to my knowledge such a self-standing "science" does not yet exist. It is like "inverse problems" a couple of decades ago: it was then just a vague term but presently it became a coherent interdisciplinary field with a learned society (AIP), international meetings, etc. It might happen to the prediction science, too. It reminds me, at a more complex level, of the math and algorithms for extrapolations and interpolations of functions and operators.

A prediction or forecast is a statement about the way things will happen in the future, often but not always based on experience or knowledge. Although guaranteed information about the future is in many cases impossible, prediction is necessary to allow plans to be made about possible developments. On the other hand, Dynamic System is a set of concepts that describe behavior as the emergent product of a self-organizing, multicomponent system evolving over time.

One can also do "predictions" about in terms of space rather than time, or other variables. Or kind of "backward predictions" regarding the past like in archeology or astronomy (where was the Moon at such and such date in distant past). That is why I prefer the term "extrapolations". In any case, before "predicting" anything regarding a beast, one must first study the beast itself, as such.

On the other hand, it is true that without doing "predictions" of some kind, Science would be useless.

Any model of phenomenons must deal with handling errors. Errors in the input data, errors in the model (which is always an approximation of reality), errors in the computations, etc. Some models are deterministic (classical mechanics), yet the errors and their propagation are never absent, so the predictive power may be limited. Some models include a stochastic component (thermodynamics, quantum mechanics) yet can lead to firm predictions in a probabilistic sense.