Why can't a critical point of a system of DEs be complex?

[zenterix]

Homework Statement

Consider the system

$$x'=x-2y+\frac{x^2}{4}$$

$$y'=5x-y-y^2$$

Relevant Equations

To find the critical points we need to find $(x,y)$ combinations that make the above equations equal to zero.

From the first equation we can write

$$y=\frac{x}{2}+\frac{x^2}{8}$$

Subbing into the rhs of the second equation and equating to zero we find (after some algebra) that

$$x(x-4)(x^2+12x+72)=0$$

This equation has roots $0$, $4$, and $-6\pm 6i$.

Then, $x=0\implies y=0$ and $x=4\implies y=4$. So $(0,0)$ and $(4,4)$ are critical points.

My question is about the cases where $x$ is a complex root.

$$x=-6+6i\implies y=-3-6i$$

$$x=-6-6i\implies y=-3+6i$$

Why aren't $(-6+6i,-3-6i)$ and $(-6-6i,-3+6i)$ considered critical points?

Note

Here is a problem set with solutions from MIT OCW's 18.03 "Differential Equations" course. The first problem asks us to find the critical points of the system above. In the solution presented, they simply say that "only the real roots give critical points".

My question is simply why this is.

 

[Orodruin]

You are solving a real differential equation on $\mathbb R^2$. The complex points are not located in $\mathbb R^2$.

 

[zenterix]

@Orodruin What is the branch of mathematics (or more practically, the typical math course) where one learns about the underlying foundations of differential equations?

 

[Orodruin]

Typically calculus (single and multi variable) followed by some dedicated courses on solving ODEs and PDEs. Linear algebra is (as always) also highly relevant.

 

[zenterix]

I meant like a pure math course.

Your answer seems to be about what is typical in most STEM fields. I just finished MIT's 18.03, and I've taken a lot of Calculus, plus linear algebra.

But sometimes I feel a large curiosity about taking some pure math courses on these topics.

I will definitely be taking two analysis courses (because I would like to know what a manifold is).

My question is in this vein. What course has the pure math foundations for differential equations?

 

[Orodruin]

What makes you think calculus is not math?

 

[pasmith]

If you are willing to accept complex valued solutions, then fixed points can obviously be nonreal.

One can always regard a system in two complex-valued unknowns as a system in four real-valued unknowns, although a system of four real unknowns admits behaviours which a system of two real unknowns does not.

 

[zenterix]

What makes you think calculus is not math?

What makes you think that I think that?

 

[Orodruin]

What makes you think that I think that?

You seem unsatisfied with it being the foundation of differential equations. You cannot get more foundational than defining the derivative operators used in the differential equations.

 

[zenterix]

You seem unsatisfied with it being the foundation of differential equations.

I am making a distinction between different kinds of calculus courses.

In particular I am making a distinction between courses that are typically called calculus and those that are called analysis.

I used the term "pure math" to denote courses in a math department that would focus on what I am calling "foundations" of what is typically taught in a calculus course (for, say, computer scientists, engineers, physicists, biologists, chemists; in other words, every other major).

I've studied basically the entirety of Spivak's Calculus and Apostol's Calculus. In 2022 I did over 600 problems in Spivak's Calculus. Still, this is not the foundations that I would find in analysis.

My question was about how I might study differential equations at the level of an analysis course. In particular, what would such a course be called? As far as I can tell, it doesn't seem to be Analysis I or Analysis II on MIT OCW.

 

[pasmith]

My question is in this vein. What course has the pure math foundations for differential equations?

The specialism is "dynamical systems". (A course describing itself as "dynamical systems and control theory" is likely to be focused on engineering applications, which is not what you are looking for.)

Some of the theory, however, fits more naturally in the realm of "functional analysis", or even basic topology and metric spaces, since one can recast the differential equation $\dot x = f(x,t)$ as the integral equation $\left[x(s)\right]_0^t = \int_0^t f(x(s),s)\,ds$; a solution of the ODE subject to $x(0) = x_0$ is then a fixed point of the map $$T: x \mapsto \left(t \mapsto x_0 + \int_0^t f(x(s),s)\,ds\right)$$ on an appropriate function space.

 

[zenterix]

Some of the theory, however, fits more naturally in the realm of "functional analysis"

Here is a course called "Introduction to Functional Analysis". Does this seem like what you are talking about?

From the names of the lectures, I can't tell what is being covered as the names are all theorems or concepts that I am unaware of.

There is Introduction to Topology.

There is also a short course called "Introduction to Metric Spaces".

 

[martinbn]

I meant like a pure math course.

Your answer seems to be about what is typical in most STEM fields.

I think that the M in STEM stands for mathematics.

 

[zenterix]

I think that the M in STEM stands for mathematics.

That's right. But let me analyze with precision what I wrote so you may understand.

I used two terms deliberately: "pure" math and "most STEM fields".

"Most STEM fields" can exclude math entirely and still be an accurate term.

In "most STEM fields" in undergraduate studies, the calculus course is most definitely not an analysis course, which is what I mean by a "pure" math course.

Here is a nice webpage from MIT's math department with the "major roadmaps" for "undergraduates interested in particular fields and applications of mathematics".

As you can see there are two categories of roadmaps: "Pure Mathematics" and "Applied Mathematics".

 

[FactChecker]

My question is about the cases where $x$ is a complex root.

$$x=-6+6i\implies y=-3-6i$$

$$x=-6-6i\implies y=-3+6i$$

Why aren't $(-6+6i,-3-6i)$ and $(-6-6i,-3+6i)$ considered critical points?
they simply say that "only the real roots give critical points".

My question is simply why this is.

That is an excellent question. In this case, it is assumed that only real values for x and y have a physical interpretation so the complex roots are not physically meaningful. If you continue farther into the subject, you will study Laplace transformations. In that, the positions of complex zeros and poles on the complex plane will tell you about the stability of the system.

 

[Orodruin]

Excluding calculus as a field from ”pure maths” is honestly ridiculous. You can have calculus courses that are more or less applied depending on who gives the course and for what audience, but it is the foundation for a lot of mathematics and in any way implying it is not ”pure” just because it has a lot of applications is bizarre.

Are you suggesting a student in ”pure maths” skip calculus?

 

[zenterix]

Excluding calculus as a field from ”pure maths” is honestly ridiculous.

I didn't write anything about what anyone should or should not do.

I also never said anything about excluding calculus from pure math.

I think you should be more careful when you read the words people write.

What I am saying is that a math major will not escape studying calculus at a more abstract level than a typical engineering or a physics student.

This is pretty obvious.

Just compare any analysis book with, to be extreme, Stewart's Calculus. Even though they go through the same topics, they are worlds apart in terms of how they go through those topics.

It's the same with linear algebra. You can take a linear algebra that is based on applications (usually with a lot of problems based on matrix manipulation), but you can also take a linear algebra course that is way more abstract (and that is probably based on theorems and proofs rather than computations).

Compare Strang's linear algebra book with Axler's. The latter makes very little use of matrices.

Use whatever alternative terms you want to but it's pretty obvious to me when I am in a course intended for math majors vs when I am in the same course intended for other majors in STEM.

I am not making a judgement about what is better or worse in general.

There is usually no need for a biologist or even a physicist to take analysis. And many mathematicians may not bother too much with applications.

As for differential equations, which is what I originally asked about, let me give you an example.

Can you prove the Poincare-Bendixson Theorem (conditions required for a limit cycle to exist) with just a regular engineering calculus course?

This is precisely the type of theorem that led me to ask my original question: how can I study differential equations at a more fundamental level?

A typical engineering student can use this theorem, but he can't prove it.

Are you suggesting a student in ”pure maths” skip calculus?

I don't know how you can even ask this question based on what has been written in this thread so far.

 

[zenterix]

If you continue farther into the subject, you will study Laplace transformations

I studied the Laplace transform but only to solve single differential equations not a system.

 

[FactChecker]

I studied the Laplace transform but only to solve single differential equations not a system.

Those basic principles are applied in very large systems. For instance, in fighter aircraft flight control development, the equations of motion and the flight control system have dozens of interrelated variables and differential equations. The stability of the entire closed-loop system is analyzed using Laplace transformations (or Z-transformations now that flight controls are digital). In fact, the flight control is diagrammed in that form and implemented in code based on the diagrams. The positions of the zeros and poles of the equations indicates how stable the system will be.

 

[Orodruin]

I think you should be more careful when you read the words people write.

Perhaps you should be more careful when writing the words, because that is exactly what it sounds like.

A typical engineering student can use this theorem, but he can't prove it.

That doesn’t mean calculus is irrelevant to it. Obviously you can teach calculus at different levels, but you asked for what branches were fundamental and calculus is:

What is the branch of mathematics (or more practically, the typical math course) where one learns about the underlying foundations of differential equations?

You then continued on to dismiss calculus as a possible answer:

I meant like a pure math course.

Which I cannot interpret in any other way than considering calculus as irrelevant.

 

[DaveE]

I think you should focus on learning what you want or need to know somewhat independently at this point. At some point it doesn't matter, or even always make sense, to ask or argue about what defines a curriculum. You can get information from lots of sources, each with their own approach. Mathematics, analysis, engineering all have many nooks and crannies to explore.

OK, back to your original question. It is often assumed, but not always explicitly stated, that DEs are constrained to real applications so $x, y \in \mathbb {R}$. Otherwise it is simply another valid solution to an equation. You, or your instructor, have to provide the meaning, the context.

Finally, I think you would like some of Steve Strogatz's lectures, particularly on dynamic systems. He does a lot of applied math.

 

[zenterix]

That doesn’t mean calculus is irrelevant to it.

The only person who used the word irrelevant in this entire thread is you. I never said anything is irrelevant.

You are defensively aggressive and have your mind set on combating a particular set of ideas that you yourself created. This is a very non-constructive approach.

you asked for what branches were fundamental

Actually I did not ask this at all.

I asked

@Orodruin What is the branch of mathematics (or more practically, the typical math course) where one learns about the underlying foundations of differential equations?

and I also then asked

What course has the pure math foundations for differential equations?

and also

My question was about how I might study differential equations at the level of an analysis course. In particular, what would such a course be called?

So, maybe you need to learn to read again.

 

[Orodruin]

So, maybe you need to learn to read again.

Resorting to ad hominem attacks does not help your case.

The only person who used the word irrelevant in this entire thread is you. I never said anything is irrelevant.

when you say:

I meant like a pure math course.

You may not be using the word ”irrelevant”, but that is how it reads: as dismissive of the answer provided to you. As if it was beneath you and not good enough. I strongly suggest that you consider how you formulate yourself - taking into account that certain formulations may read very differently from how you have intended. You also need to take this into account when reading answers.

 

[zenterix]

I think you should focus on learning what you want or need to know somewhat independently at this point.

I agree and that is why I asked for where I might find the next steps now that I completed an entire course from MIT on differential equations. I measure the time I dedicate to each activity. I spent about 300h on the course: lectures, problem sets, recitations, reading assigned chapters and notes, doing extra problems, and doing the exams.

I can compute answers to problems using various techniques and I can model simple problems.

One avenue I will pursue, naturally, is the application of what I have learned in other courses. In particular, the course 8.03 "Waves and Vibrations" and after that 8.04 "Quantum Physics I".

But as I stated previously in this thread, I am curious to learn about differential equations also from the point of view of the underlying theorems and their proofs.

Will check out Steven Strogatz. I see he has a course on Nonlinear Dynamics and Chaos on Youtube.

 

[zenterix]

For the record, I am done with this thread, and will not answer any other further posts.

 

[Orodruin]

As I said, you are defensive

Pot calling the kettle black if I ever saw it.

Nope, that is how YOU choose to read it.

My point exactly, which you refuse to understand. If you post something that can be interpreted in several ways. Someone will interpret it in a way you may have not intended. It is up to you to communicate it unambiguously.

I am looking for a pure mathematics course, as opposed to an applied mathematics course.

You are also wrong in your assertion that STEM programs other than mathematics will not have pure maths courses in them. This may or may not be in different programs. Case in point: My undergrad was in engineering physics. My math courses were taught by the math department focusing on the math rather than on applications. I have taken numerous courses you would classify as ”pure math” as part of my undergrad.

So yes, dismissed your notion that calculus and linear algebra are enough to achieve my goals.

Then that is what you should have asked, which you did not. You should have been more specific in your question. If someone takes the time to answer your question and it did not come across as you intended, then it is up to you to clarify without being dismissive and condescending because you were not clear enough.

 

[berkeman]

Thread is paused for Mentor review...

 

[PeterDonis]

I am done with this thread, and will not answer any other further posts.

Which is a good reason (but hardly the only one) to leave the thread closed. So we will.