That's because complex structures, not just in the complex plane but on general complex geometries, are combinations of the "symplectic structure" which is used here to divide coordinates into position and momentum phase spaces and metric structures, ie dot products. It just turns out you do not need the dot product to take the index (symplectic spaces are all even dimensional like complex spaces, and complex maps can be seen as vector fields just as they are here - hamiltonian vector fields). This is realized more abstractly in "kahler geometry," but you can explore the index ideas here further in the work of the classical mechanist Vladimir Arnold, he has good books on differential equations and mechanics.
Sir I want to see difference between 1)linear - non linear slope fields 2)1st order- 2nd order slope fields 3)1st degree - 2nd degree slope fields 4)homogeneous - Non-Homogeneous slope fields. I can't find them anywhere... Can you please help??
Maybe you could find this interesting: recently I have figure out that no linear differential equation solution could achieve by itself the value zero and stays there forever (will violate the conditions for uniqueness), so I have been trying to find a physical model that could have a finite extinction time were the dynamics dies. So far I have made this modified version of the nonlinear damped pendulum with a sublinear friction term for some positive constants a and b: x'' + a*sgn(x')*sqrt(|x'|)*(sqrt(2)/4+|x'|^(3/2))+b*sin(x)=0, x(0)=pi/2, x'(0)=0 But I have not being able to prove that they indeed have a finite ending time. Do you know if its plausible to prove it through modern nonlinear analysis tools?
@@whatitmeans Re: "I have been trying to find a physical model that could have a finite extinction time were the dynamics dies": I'm not sure I follow. How about the motion of a brick on a horizontal surface with constant friction between the brick and the surface? Assuming the brick moves along the surface at t=0, it'll stop eventually. Newton's law: x"=-k*g*sign(x'), where g is 9.8m/s^2 and k is the friction coef.
@@siguc Thanks, I have look for those examples a lot without finding any: the nearest thing was the Norton's Dome example (which I don't know if its a reallistic example), and the Coulomb Force... Does that kind of system have an specific name?(that maybe I am missing), Do you have any source for them?
@@siguc I have searched for your example on google but I didn't find much info so I hope you could share some sources to it... Previously I have found that: x'=-sgn(x)*sqrt(|x|), x(0)=1 can stand the solution: x(t)=1/4*(1-t/2+|1-t/2|)^2 which has t=2 as ending time. For your example, asuming that all constants equals one, and with initial conditions y(0)=2 and y'(0)=-2 (so, the brick starts instantly reducing the speed after a kick with |y'|=2), in a similar way (through trial and error in Wolfram-Alpha), I think y(t)=1/2*(1-t/2+|1-t/2|)^2 is a solution for your example y''=-sgn(y'), y(0)=2, y'(0)=-2 Achieving and ending time at t=2. Hope you could share any source to equations with their solutions, so far the only thing I found is: en.m.wikipedia.org/wiki/Contact_dynamics#Non-smooth_approach Thank you very much.
Just right perfect elegant articulation of complex concepts in field of non linear dynamics applied mathematics Modelled on Quantum information Processing universe visualised both waveforms and particles simultaneously unfolding hidden connections, patterns and flows States Chained Phases Transitions Surfacing Planes Traced Source Energy Packets Collapsed TRUTHS functions Vectors Scalars Spacetime Ordered Resonant Frequency Following Responses Induced real world controlled Eco-Systems.
See this shit? THIS is why I love math. You tackle one topic, then months later you learn something else and you see these huge connections. This was screaming residue theorem from the moment you had the 1/2pi integral!
I would have a few distinct questions: - How does this scale to higher dimensional spaces? - Is this computationally feasible? - What if there is some uncertainty in the parameters of the differential equation? - In particular, when varying the parameters of the DE, the closed curves should appear / disappear. How robust is this method with respect to the uncertainty of the parameters? Ideally, one would like some computationally feasible, quantifiable estimate of "how far" the DE is away from one with/without a closed orbit. Is that feasible at all?
Isn't this just Stokes' theorem, saying the curl is the integral of the divergence of the path from the vector field? I.e., orbits require integer curls, and stable orbits require positive curls. If so it seems like this can be mapped to just a statement about having zero net curvature for the path in the field, which would be pretty obvious for physical trajectories -- if an orbit experiences a non-zero force integration within one period, it will not follow the same path again, i.e., is an unstable orbit.
It very much is a topological invariant! As we're only interested in vectors' directions, we are replacing nonzero vectors with unit vectors (mapping to the "unit tangent bundle"). Since we're in the plane, the space of unit vectors is S^1 (a topological circle). The space of loops in phase space is also S^1, so we're thinking about maps from S^1 -> S^1. The *index* of a fixpoint is the *degree* of this map from S^1 -> S^1, and the degree is a topological invariant.
vibes of gauss law in electromagnetism, where the surface integral of the field is proportional to the sum of charges (which as a bonus are integer because of quantum mechanics)
Is there a problem with your development of "Fixed Point" for I at about 7:50? I ask this because the expression inside the integral is 0/0 (zero divided by zero) . No! You have made your point with a bit of drama that the index is undefined for fixed points. Yes? Brilliantly done. Thank you!
Hi, thanks for the question. Fixed points (x*,y*) are defined as the points where dx/dt = dy/dt = 0. If we plug a fixed point into the expression for the Index (the integral eqn) then we will get an expression divided by 0 which is undefined. This means the index is undefined at fixed points. If you want to learn more about fixed points I define them from scratch in this video: ua-cam.com/video/6mLCFyEv3Z0/v-deo.html
I suspect the sign of the index is connected to the sign of the Gaussian curvature of the surface generated by integrating the vectors as if they were slopes on a surface. (Think of the vectors as water flow on the surface of a mountain ridge to grok this.)
Hi, I just managed to squeeze this video out before work got too overwhelming haha. I don't think I'll be able to produce another video until early next year sadly. The next video will likely be about limit cycles. Although, I might take a different path and talk about Floquet theory. Unsure at this point :)
Recently I found that every differential equation that have as solution a non-piecewise Power Series can have only never-ending solutions in time, and even worst, if a solution has a finite extinction time, then the differential equation cannot fulfill uniqueness of solutions! (it does only before the ending time). So all things we were teached on physics cannot accurately represent a phenomena with dynamics that by itself becomes zero and stays there forever (it violates the Identity Theorem for Power Series). So, those sinks point aren't really sinks, the curves will never reach those points in a finite time. But this doesn't mean that they cannot be modeled, but instead that maybe we have be using the mistaken tools: as a simple example, think in the following equation x'=-sgn(x)*sqrt(|x|), x(0)=1 which can stand the solution x(t) = 1/4*(1-t/2+|1-t/2|)^2 which becomes zero at t=2 and stays there forever. But it also stands solutions that rises from zero at t=2, which could represent the same kind of issues as the Norton's Dome example, but fortunately between initial conditions and final time t=2 the uniqueness is granted. It could be interesting to see some analysis of the nonlinear dynamics of these kind of finite time systems, as example, think in this modified version of the nonlinear damped pendulum with a sublinear friction term for some positive constants a and b: x'' + a*sgn(x')*sqrt(|x'|)*(sqrt(2)/4+|x'|^(3/2))+b*sin(x)=0, x(0)=pi/2, x'(0)=0 How are its dynamics? It really exists a finite extinction time T_f where it becomes zero forever after? It exist a time T_n where the solution stops oscillating? (since it is going to become zero it should exist) Hope you could use this example to your video series and analyse its behavior with current tools for nonlinear dynamics.
Sir I want to see difference between 1)linear - non linear slope fields 2)1st order- 2nd order slope fields 3)1st degree - 2nd degree slope fields 4)homogeneous - Non-Homogeneous slope fields. I can't find them anywhere... Can you please help??
That seems ok in 2D, but ... imagine an 8 curve, but in 3D, where the middle "intersection" point is at different level (z). We have a close orbit, but isn't it that the index is ... eventually, two? (inspired from some "orbit" of an electron). Does the "index" works only with 2 variables?
Hey Jude, good question. Any simple closed curve that surrounds only the saddle in the middle will have an index of -1. This can also be seen at 5:20. Because of this fact, it's impossible for a closed orbit (ie a periodic solution) to exist that surrounds only the saddle fixed point. This is because a closed orbit can only exist if it has an index of +1.
I'm unsure if this is the basis of topology (I doubt it), but some of the ideas expressed in this video at 8:47 use formulas from topology. In particular we make use of a homotopy parameter to describe the "deformation" of the curve.
What a great question! Sadly, I'm not an electrical engineer so I can't answer that question definitively. However, I wouldn't be surprised if nonlinear dynamics plays a large role in this field (at least for specific applications). For what it's worth, the Van der Pol oscillator was originally a model for a certain electrical circuit and can be viewed using an oscilloscope (I found this video here showing that ua-cam.com/video/ZeIxnYLHJuo/v-deo.html)
Hi, thanks for the question. What do you mean by sampling? The index can easily be calculated by adding the indices of the fixed points inside the curve.
@@virtually_passed Sure, but say I don't know what the fixed points are, and I want to calculate the index, how's that done? By sampling I meant numerically sampling points along a curve to estimate the index. Thanks for the reply by the way : )
@@user-sl6gn1ss8p As far as I know there are only 4 ways: 1) Find the fixed points inside the curve (by solving for dx/dt = dy/dt = 0) and characterize them from their eigenvalues and then determine their index. (this is probably the easiest way) 2) Calculate the index analytically using the integral expressed at 7:12 3) Calculate the index numerically using the same integral at 7:12. This can be done using a riemann integral. 4) 'Sampling' phi at various points and counting the rotations. I'd recommend option 1. Failing that I'd recommend option 3. Hope that helps. I'd strongly advise not doing 4 because near a fixed point you can get all types of numerical rounding that could mess up your measurement.
That's a really interesting question! Let's consider dx/dt = y dy/dt = 0 There is a line of fixed points along the x axis (because y = 0 satisfies dx/dt = dy/dt = 0) In this case, you can't draw a simple closed curve that goes through the x axis because the curve cannot pass through a fixed point (because phi is undefined at a fixed point). So the index will = 0 for any other curve you draw. As a result, you can say this system will never have any periodic orbits.
Another thing, everytime I see the thumbnail of the 2nd video of this series, I imagine if it's possible to calculate things like the time it takes to make 3 revolutions (like back and forth), or calculate how many revolutions until it stops or some other calculations. Is it possible? I imagine it being useful as an applied maths exercise
Hi :) I assume you mean the thumbnail for this video, right? ua-cam.com/video/I9UEBRya4X0/v-deo.html&ab_channel=VirtuallyPassed This is for a spring mass damper system. This spring mass damper system is very nice because it's fully linear (m,c,k are all constant) so it's possible to derive an expression for its motion analytically with an equation! You can derive from first principles that the frequency of this system is: f = w_n*√(1-ζ^2)/(2*π) Where w_n = √(k/m) And 2*ζ*w_n = c/m This means that the time it takes to make 3 (incomplete) revolutions is: T = 3 / f ----------- in case you're interested ----------- It's comparably quite easy to find the period of periodic orbits for linear systems. For nonlinear systems its waaay harder. In fact, just proving that periodic orbits exist is quite hard for nonlinear systems.
@@virtually_passed yeah that's the one I meant! I knew about that dynamic system (not the actual formulas you mentioned) but wanted to know if there were some approaches in general to other dynamical systems. Now I know they are difficult lol. But hey! That could be a great idea for a video if you find examples for it!
@@joeeeee8738 A crazy fact about nonlinear systems is that some of them generally have no analytical solution! That means it's impossible to get an equation to predict their motion using 'normal' operators like sqrt(), exp(), +, -, /, * . For this reason, a lot more emphasis is placed on finding out whether a periodic orbit exists or not, and how stable it is, rather than finding an equation that determines its exact period. That being said, there is a method called "Harmonic Balancing" which can roughly approximate the period of a periodic solution.
Bro he is not a scientist that he will teach us his own theory. Every teacher do the same,they help us to understand things that's difficult to study by ourselves.
Hi, I've already made a video about linear 2nd order differential equations & Vector Fields here: ua-cam.com/video/I9UEBRya4X0/v-deo.html and an intro to nonlinear 2nd order differential equations & Vector Fields here: ua-cam.com/video/6mLCFyEv3Z0/v-deo.html Is that what you're looking for? :)
These ideas give vibes of complex residue theorem
That's because complex structures, not just in the complex plane but on general complex geometries, are combinations of the "symplectic structure" which is used here to divide coordinates into position and momentum phase spaces and metric structures, ie dot products. It just turns out you do not need the dot product to take the index (symplectic spaces are all even dimensional like complex spaces, and complex maps can be seen as vector fields just as they are here - hamiltonian vector fields). This is realized more abstractly in "kahler geometry," but you can explore the index ideas here further in the work of the classical mechanist Vladimir Arnold, he has good books on differential equations and mechanics.
@@nicholasandrzejkiewicz woah
I thought "I forgot much of complex analysis, I guess I won't notice that". And then I noticed it lol
Ah yes yes nerd Group Theorums .. i feel at🏡
I already explained this on my chan with less jargon and more prop
I find it very very amizing series that talk on some concepts
associated with clair explanations.
thank you so much.
As a PhD working with nonlinear dynamics, this video was very entertaining to watch!
Sir I want to see difference between
1)linear - non linear slope fields
2)1st order- 2nd order slope fields
3)1st degree - 2nd degree slope fields
4)homogeneous - Non-Homogeneous slope fields.
I can't find them anywhere...
Can you please help??
Maybe you could find this interesting: recently I have figure out that no linear differential equation solution could achieve by itself the value zero and stays there forever (will violate the conditions for uniqueness), so I have been trying to find a physical model that could have a finite extinction time were the dynamics dies.
So far I have made this modified version of the nonlinear damped pendulum with a sublinear friction term for some positive constants a and b:
x'' + a*sgn(x')*sqrt(|x'|)*(sqrt(2)/4+|x'|^(3/2))+b*sin(x)=0, x(0)=pi/2, x'(0)=0
But I have not being able to prove that they indeed have a finite ending time.
Do you know if its plausible to prove it through modern nonlinear analysis tools?
@@whatitmeans Re: "I have been trying to find a physical model that could have a finite extinction time were the dynamics dies": I'm not sure I follow. How about the motion of a brick on a horizontal surface with constant friction between the brick and the surface? Assuming the brick moves along the surface at t=0, it'll stop eventually. Newton's law: x"=-k*g*sign(x'), where g is 9.8m/s^2 and k is the friction coef.
@@siguc Thanks, I have look for those examples a lot without finding any: the nearest thing was the Norton's Dome example (which I don't know if its a reallistic example), and the Coulomb Force... Does that kind of system have an specific name?(that maybe I am missing), Do you have any source for them?
@@siguc I have searched for your example on google but I didn't find much info so I hope you could share some sources to it...
Previously I have found that:
x'=-sgn(x)*sqrt(|x|), x(0)=1
can stand the solution:
x(t)=1/4*(1-t/2+|1-t/2|)^2
which has t=2 as ending time.
For your example, asuming that all constants equals one, and with initial conditions y(0)=2 and y'(0)=-2 (so, the brick starts instantly reducing the speed after a kick with |y'|=2), in a similar way (through trial and error in Wolfram-Alpha), I think
y(t)=1/2*(1-t/2+|1-t/2|)^2
is a solution for your example
y''=-sgn(y'), y(0)=2, y'(0)=-2
Achieving and ending time at t=2.
Hope you could share any source to equations with their solutions, so far the only thing I found is: en.m.wikipedia.org/wiki/Contact_dynamics#Non-smooth_approach
Thank you very much.
As an undergrad taking a nonlinear dynamics class this semester, these videos are super relevant and clearly explained. Great job!
thank you for choosing that pfp. always a joy to see Yoshi commenting on an ODE video
Extremely underrated channel!!!
Kep doing ao and channel will grow to millions :)❤
Thank you for marvelous series. Visual explanations are on spot!
A grade stuff mate, some serious talent here! Subscribed instantly!
cheers, mate :)
What a brilliant video! An incredible explanation with really really good visualizations. Keep up the good work!
Thanks!
Waiting eagerly for the next video. Cheers
Next video will release early next year hopefully
I honestly was super lost halfway through but the conclusion summed it up really well that it instantly all became clear again.
This appears to me to be quite related to complex analysis and poles... would be interesting to know how.
It is, it's generalization of the specific result you have in complex analysis, search for winding number in Wikipedia.
just wow..amazing..keep uploading such stuff..it's very helpful
Thanks! I'm working on more videos now 🙂
Very interesting, informative and worthwhile video.
Thanks!
Wish I had seen this video when I was learning Field Theory!
Amazing series! Can't wait for the next video
Just right perfect elegant articulation of complex concepts in field of non linear dynamics applied mathematics Modelled on Quantum information Processing universe visualised both waveforms and particles simultaneously unfolding hidden connections, patterns and flows States Chained Phases Transitions Surfacing Planes Traced Source Energy Packets Collapsed TRUTHS functions Vectors Scalars Spacetime Ordered Resonant Frequency Following Responses Induced real world controlled Eco-Systems.
See this shit? THIS is why I love math. You tackle one topic, then months later you learn something else and you see these huge connections. This was screaming residue theorem from the moment you had the 1/2pi integral!
I would have a few distinct questions:
- How does this scale to higher dimensional spaces?
- Is this computationally feasible?
- What if there is some uncertainty in the parameters of the differential equation?
- In particular, when varying the parameters of the DE, the closed curves should appear / disappear. How robust is this method with respect to the uncertainty of the parameters?
Ideally, one would like some computationally feasible, quantifiable estimate of "how far" the DE is away from one with/without a closed orbit. Is that feasible at all?
Isn't this just Stokes' theorem, saying the curl is the integral of the divergence of the path from the vector field? I.e., orbits require integer curls, and stable orbits require positive curls. If so it seems like this can be mapped to just a statement about having zero net curvature for the path in the field, which would be pretty obvious for physical trajectories -- if an orbit experiences a non-zero force integration within one period, it will not follow the same path again, i.e., is an unstable orbit.
If only I could have had you as a math teacher, _I could've been somebody._
Awesome video!
Great explanations and visualizations
Thanks so much :)
Great one, the whole thing reminded me of topological invariants and curvatures of fields
It very much is a topological invariant! As we're only interested in vectors' directions, we are replacing nonzero vectors with unit vectors (mapping to the "unit tangent bundle"). Since we're in the plane, the space of unit vectors is S^1 (a topological circle). The space of loops in phase space is also S^1, so we're thinking about maps from S^1 -> S^1. The *index* of a fixpoint is the *degree* of this map from S^1 -> S^1, and the degree is a topological invariant.
vibes of gauss law in electromagnetism, where the surface integral of the field is proportional to the sum of charges (which as a bonus are integer because of quantum mechanics)
I'm looking forward to the next video
Is there a problem with your development of "Fixed Point" for I at about 7:50? I ask this because the expression inside the integral is 0/0 (zero divided by zero) . No! You have made your point with a bit of drama that the index is undefined for fixed points. Yes? Brilliantly done. Thank you!
Hi, thanks for the question. Fixed points (x*,y*) are defined as the points where dx/dt = dy/dt = 0. If we plug a fixed point into the expression for the Index (the integral eqn) then we will get an expression divided by 0 which is undefined. This means the index is undefined at fixed points. If you want to learn more about fixed points I define them from scratch in this video:
ua-cam.com/video/6mLCFyEv3Z0/v-deo.html
Thank you so much
That was awesome
Interesting! 🤔
Very well done video
Thanks!
I suspect the sign of the index is connected to the sign of the Gaussian curvature of the surface generated by integrating the vectors as if they were slopes on a surface. (Think of the vectors as water flow on the surface of a mountain ridge to grok this.)
Great! Can't wait for the next episode? When is it coming?
Hi, I just managed to squeeze this video out before work got too overwhelming haha. I don't think I'll be able to produce another video until early next year sadly. The next video will likely be about limit cycles. Although, I might take a different path and talk about Floquet theory. Unsure at this point :)
Wow! Awesome! Thanks
As someone with no PhD, no masters, or bachelors, i say this is good. If you don’t believe me then just ask me.
simple yet cool
the eerie connection between this and complex residue means there must be a deeper connection
I sort of wish 3b1b’s videos looked this good
Thanks but I reckon 3b1b has me beat 100% :)
It's just a hunch, but is this related to the residue theorem?
Recently I found that every differential equation that have as solution a non-piecewise Power Series can have only never-ending solutions in time, and even worst, if a solution has a finite extinction time, then the differential equation cannot fulfill uniqueness of solutions! (it does only before the ending time).
So all things we were teached on physics cannot accurately represent a phenomena with dynamics that by itself becomes zero and stays there forever (it violates the Identity Theorem for Power Series).
So, those sinks point aren't really sinks, the curves will never reach those points in a finite time.
But this doesn't mean that they cannot be modeled, but instead that maybe we have be using the mistaken tools: as a simple example, think in the following equation
x'=-sgn(x)*sqrt(|x|), x(0)=1
which can stand the solution
x(t) = 1/4*(1-t/2+|1-t/2|)^2
which becomes zero at t=2 and stays there forever.
But it also stands solutions that rises from zero at t=2, which could represent the same kind of issues as the Norton's Dome example, but fortunately between initial conditions and final time t=2 the uniqueness is granted.
It could be interesting to see some analysis of the nonlinear dynamics of these kind of finite time systems, as example, think in this modified version of the nonlinear damped pendulum with a sublinear friction term for some positive constants a and b:
x'' + a*sgn(x')*sqrt(|x'|)*(sqrt(2)/4+|x'|^(3/2))+b*sin(x)=0, x(0)=pi/2, x'(0)=0
How are its dynamics?
It really exists a finite extinction time T_f where it becomes zero forever after?
It exist a time T_n where the solution stops oscillating? (since it is going to become zero it should exist)
Hope you could use this example to your video series and analyse its behavior with current tools for nonlinear dynamics.
Sir I want to see difference between
1)linear - non linear slope fields
2)1st order- 2nd order slope fields
3)1st degree - 2nd degree slope fields
4)homogeneous - Non-Homogeneous slope fields.
I can't find them anywhere...
Can you please help??
Is this a sort of dynamical interpretation of the residue theorem?
Beautiful!
Great video as always thank you.
Which software or library do you use to generate and simulate these animations?
Thanks :) I coded each of these simulations using Python. In particular, I used a library called "manim". Good luck!
Gives very strong complex analysis vibes
Make a video on method of characteristics, partial differential equation
That seems ok in 2D, but ... imagine an 8 curve, but in 3D, where the middle "intersection" point is at different level (z). We have a close orbit, but isn't it that the index is ... eventually, two? (inspired from some "orbit" of an electron). Does the "index" works only with 2 variables?
Last semester I took an ODE course where we were introduced to Lyapunov stability. Are there connections between this and Index Theory?
As far as I know, there are no connections between Lyapunov stability and Index theory :)
Good shit man
Thanks man
Do the equations you listed for cell division ODEs have a name?
At 13:31, what if you drew a closed orbit around the saddle in the middle? Would the index also be equal to +1?
Hey Jude, good question. Any simple closed curve that surrounds only the saddle in the middle will have an index of -1. This can also be seen at 5:20.
Because of this fact, it's impossible for a closed orbit (ie a periodic solution) to exist that surrounds only the saddle fixed point. This is because a closed orbit can only exist if it has an index of +1.
Nice video
Is this the basis for topology?
I'm unsure if this is the basis of topology (I doubt it), but some of the ideas expressed in this video at 8:47 use formulas from topology. In particular we make use of a homotopy parameter to describe the "deformation" of the curve.
this is very very very similar to the residue theorem!
Does this relate to electrical engineering?
What a great question! Sadly, I'm not an electrical engineer so I can't answer that question definitively. However, I wouldn't be surprised if nonlinear dynamics plays a large role in this field (at least for specific applications). For what it's worth, the Van der Pol oscillator was originally a model for a certain electrical circuit and can be viewed using an oscilloscope (I found this video here showing that ua-cam.com/video/ZeIxnYLHJuo/v-deo.html)
What happens in 3 dimensions, N dimensions?
that's a really great question! I've never heard of this theory being extended to 3D so I'm unsure, sorry!
Does that mean if I slap you multiple times then this can predict at what time my hand will touch your face?
That's freakin amazing.
yep, that's pretty much it lol :P
Some of these closed curves simular to hysteris loop
Hello there! Can you give me some books on the topic?
I use Nonlinear Dynamics and Chaos by Steven H strogatz
Any (general) way to calculate the indexes other than by sampling?
Hi, thanks for the question. What do you mean by sampling? The index can easily be calculated by adding the indices of the fixed points inside the curve.
@@virtually_passed Sure, but say I don't know what the fixed points are, and I want to calculate the index, how's that done?
By sampling I meant numerically sampling points along a curve to estimate the index.
Thanks for the reply by the way : )
@@user-sl6gn1ss8p As far as I know there are only 4 ways:
1) Find the fixed points inside the curve (by solving for dx/dt = dy/dt = 0) and characterize them from their eigenvalues and then determine their index. (this is probably the easiest way)
2) Calculate the index analytically using the integral expressed at 7:12
3) Calculate the index numerically using the same integral at 7:12. This can be done using a riemann integral.
4) 'Sampling' phi at various points and counting the rotations.
I'd recommend option 1. Failing that I'd recommend option 3. Hope that helps. I'd strongly advise not doing 4 because near a fixed point you can get all types of numerical rounding that could mess up your measurement.
@@virtually_passed makes sense. Thanks : )
Do you mind if I recommend this Video in my class?
That's fine :)
When the next episode appears?
I plan to make an entire series consisting of 6 more videos this year. The next one won't be released for a while though. Likely in 2-3 months.
but what about a line of fixed points?
That's a really interesting question! Let's consider
dx/dt = y
dy/dt = 0
There is a line of fixed points along the x axis (because y = 0 satisfies dx/dt = dy/dt = 0)
In this case, you can't draw a simple closed curve that goes through the x axis because the curve cannot pass through a fixed point (because phi is undefined at a fixed point). So the index will = 0 for any other curve you draw. As a result, you can say this system will never have any periodic orbits.
14:08 More like: It's impossible for a closed orbit to exist here which encircles these poles.
Another thing, everytime I see the thumbnail of the 2nd video of this series, I imagine if it's possible to calculate things like the time it takes to make 3 revolutions (like back and forth), or calculate how many revolutions until it stops or some other calculations. Is it possible? I imagine it being useful as an applied maths exercise
Hi :) I assume you mean the thumbnail for this video, right?
ua-cam.com/video/I9UEBRya4X0/v-deo.html&ab_channel=VirtuallyPassed
This is for a spring mass damper system. This spring mass damper system is very nice because it's fully linear (m,c,k are all constant) so it's possible to derive an expression for its motion analytically with an equation!
You can derive from first principles that the frequency of this system is:
f = w_n*√(1-ζ^2)/(2*π)
Where w_n = √(k/m)
And 2*ζ*w_n = c/m
This means that the time it takes to make 3 (incomplete) revolutions is:
T = 3 / f
----------- in case you're interested -----------
It's comparably quite easy to find the period of periodic orbits for linear systems. For nonlinear systems its waaay harder. In fact, just proving that periodic orbits exist is quite hard for nonlinear systems.
@@virtually_passed yeah that's the one I meant! I knew about that dynamic system (not the actual formulas you mentioned) but wanted to know if there were some approaches in general to other dynamical systems. Now I know they are difficult lol. But hey! That could be a great idea for a video if you find examples for it!
@@joeeeee8738 A crazy fact about nonlinear systems is that some of them generally have no analytical solution! That means it's impossible to get an equation to predict their motion using 'normal' operators like sqrt(), exp(), +, -, /, * . For this reason, a lot more emphasis is placed on finding out whether a periodic orbit exists or not, and how stable it is, rather than finding an equation that determines its exact period. That being said, there is a method called "Harmonic Balancing" which can roughly approximate the period of a periodic solution.
Next video is now available! ua-cam.com/video/-I5VPE1Hbwg/v-deo.html
3-Body Systems have entered chat...
I thought you had a million subscribers
Nope, waaay less haha
WTH DOES THIS ALWAYS WORK!?
For 2D, yep.
Don't call us " guys". Just say hello, for God's sake.
Wot
bro taking strogatz notes and putting them into video format is not education
Bro he is not a scientist that he will teach us his own theory. Every teacher do the same,they help us to understand things that's difficult to study by ourselves.
Sir please make video to differentiate between linear and nonlinear direction fields.
Hi, I've already made a video about linear 2nd order differential equations & Vector Fields here: ua-cam.com/video/I9UEBRya4X0/v-deo.html
and an intro to nonlinear 2nd order differential equations & Vector Fields here: ua-cam.com/video/6mLCFyEv3Z0/v-deo.html
Is that what you're looking for? :)