- Captured On
- Source
- Limitations of the Linear: Limit Cycles and Chaos | Unit IV: First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare
1 What is the form of the solutions that trace out a closed curve?
1.1 Front
What is the form of the solutions that trace out a closed curve?
In a non-linear system
1.2 Back
The solution \(\vb{x}(t)\) will be geometrically realized by a point which goes round and round the curve \(C\) with a certain period \(T\)
\(\vb{x}(t) = (x(t), y(t))\)
will be a pair of periodic functions with period \(T\)
\(x(t + T) = x(t)\), \(y(t + T) = y(t)\) for all \(t\)
2 How are the trajectories nearby a stable limit cycle?
2.1 Front
How are the trajectories nearby a stable limit cycle?
2.2 Back
Nearby curves spiral towards \(C\) on both sides, but never will touch it.
3 How are the trajectories nearby an unstable limit cycle?
3.1 Front
How is are trajectories nearby an unstable limit cycle?
3.2 Back
Nearby curves spiral away from \(C\) on both sides, but never will touch it.
4 How are the trajectories nearby a semi-stable limit cycle?
4.1 Front
How are the trajectories nearby a semi-stable limit cycle?
4.2 Back
The trajectories outside the close curve \(C\) towards to \(C\), but trajectories inside the curve will go away from \(C\) to a critical points. The trajectories never touches each other.
5 How are the trajectories nearby a neutrally stable center?
5.1 Front
How are the trajectories nearby a neutrally stable center?
5.2 Back
6 Can a trajectory touch a limit cycle?
6.1 Front
Can a trajectory touch a limit cycle?
6.2 Back
No, 2 solutions can’t touch each other. It’s the sketch principle based on integrate curves.
7 Is it easy to find limit cycles if they exits?
7.1 Front
Is it easy to find limit cycles if they exits?
7.2 Back
No, little is known about how to do this, or how to show that a system has no limit cycles. There is active research in this subject today (02/2020)
You can use a computer for search them, but you need to focus where are you looking at.
8 What is the name of the theorem to check if there is a stable limit cycle?
8.1 Front
What is the name of the theorem to check if there is a stable limit cycle?
The main tool which historically has been used.
8.2 Back
Poincare-Bedixson Theorem
- Suppose \(R\) is the finite region between 2 simply closed curves
- \(\vb{F}\) is the velocity vector of the system
\(R\) no contains critical points and the velocity vector show like this
It’s difficult to use it.
9 When can we say that there is no closed trajectories inside a region
9.1 Front
When can we say that there is no closed trajectories inside a region
For the system
- \(\dot{x} = f(x,y)\)
- \(\dot{y} = g(x,y)\)
9.2 Back
Using the Bendixson’s Criterion, if \(f_x\) and \(g_y\) are continuous in a region \(R\) which is simply-connected (i.e. without holes) and
\({\displaystyle \pdv{f}{x} + \pdv{g}{y} \neq 0}\) at any point of \(R\)
then the system
\({\displaystyle \begin{matrix} \dot{x} &= f(x,y) \\ \dot{y} &= g(x,y) \end{matrix}}\)
has no closed trajectories inside \(R\)
10 How is the flux across closed trajectory for this system?
10.1 Front
How is the flux across closed trajectory for this system?
- \(\dot{x} = f(x,y)\)
- \(\dot{y} = g(x,y)\)
10.2 Back
The velocity field for this system is composed by \(\dot{x}\) and \(\dot{y}\), \(\vb{F} = \ev{f(x,y), g(x,y)}\)
\({\displaystyle \oint_{C} \vb{F} \cdot \vb{n} \dd{s} = 0 }\)
Because along the closed trajectory (which is solution of the system), \(\vb{n} \dd{s}\) it’s perpendicular to the velocity vector defined by \(\vb{F}\), so the integrand is always \(0\)
11 What is the Bendixson’s Criterion
11.1 Front
What is the Bendixson’s Criterion
- \(\dot{x} = f(x,y)\)
- \(\dot{y} = g(x,y)\)
11.2 Back
As \(\vb{F} = \ev{f(x,y), g(x,y)}\) is the velocity field continuously differentiable, and \(R\) is a simply-connected region.
If \({\displaystyle \div{\vb{F}} = \pdv{f}{x} + \pdv{g}{y} \neq 0}\) at any point of \(R\), then the system has no closed trajectories inside \(R\)
12 What does it happen if the region is simply-connected and there isn’t any critical point?
12.1 Front
What does it happen if the region is simply-connected and there isn’t any critical point?
Are there closed trajectories in this region \(R\)?
12.2 Back
No, this is the Critical-point Criterion
Instead, if there is a critical point in this region, this criterion say that there is a limit cycle. But using the Poincare-Bendixson theorem could not be a critical point inside of its region. The difference is that these latter regions always contain a hole
13 For what values of a and d, have closed trajectories?
13.1 Front
For what values of a and d, have closed trajectories?
- \(\dot{x} = ax + by\)
- \(\dot{y} = cx + dy\)
13.2 Back
By Bendixson’s criterion, \(a + d \neq 0 \implies\) no closed trajectories
If \(a + d = 0\), Bendixson criterion says nothing.
We go back to our analysis of the linear system. The characteristic equation of the system is
\({\displaystyle \lambda^2 - (a + d)\lambda + (ad - cd) = 0}\)
Assume \(a + d = 0\). Then the characteristic roots have opposite sign if \(ad - bc \lt 0\) and the system is a saddle; the roots are pure imaginary if \(ad - bc \gt 0\) and the system is a center, which has closed trajectories. Thus
The system has closed trajectories \(\Leftrightarrow a + d = 0, ad - bc \gt 0\)