Lectures on Nonlinear Dynamical Systems 

Here is a link to my lectures on nonlinear dynamical systems given at York University during the Winter semester of 2017. 

These lectures start off with manifold theory, and end with examples in biology, game theory, and general relativity/cosmology. 

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Dynamical Systems in Cosmology Lectures

In the final two lectures of my differential equations class , I discussed how Dynamical Systems theory can be used to understand and describe the dynamics of cosmological solutions to Einstein’s field equations. Videos and lecture notes posted below:

Lecture Notes:





Mathematics Behind The Triangle Offense

It was pointed out to me recently that a few of the articles I have written describing the detailed geometric structure behind the triangle offense is scattered in various places around my blog, so here is a list of the articles in one convenient place: 

  • The Mathematics of Filling the Triangle (First article) 
  • Group Theory and Dynamical Systems Theory Behind The Triangle Offense 
  • A Demonstration That The Triangle Offense is the most efficient/optimal way for 5 players to space the floor.
  • By: Dr. Ikjyot Singh Kohli (About the Author)

    1. The Mathematics of Filling the Triangle (First article) 
    2. Group Theory and Dynamical Systems Theory Behind The Triangle Offense 
    3. A Demonstration That The Triangle Offense is the most efficient/optimal way for 5 players to space the floor.

    By: Dr. Ikjyot Singh Kohli (About the Author)

    The Mathematics of The Triangle Offense, Continued…

    In a previous post, I showed how given random positions of 5 players on the court that they could “fill” the triangle. The main geometric constraint is that 5 players can form 3 triangles on the court, and that due to spacing requirements, these triangles are “optimal” if they are equilateral triangles.

    Given that we now know how to fill the triangle, the question that this post tries to address is that how can players actually move within the triangle. The key is symmetry. Players must all move in a way such that the equilateral triangles remain invariant. Equilateral triangles have associated with them the D_{3} dihedral symmetry group. They are therefore invariant with respect to 120 degree rotations, 240 degree rotations, 0 degree rotations, and three reflections.

    There are therefore six generators of this group:
    \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right), \left( \begin{array}{cc} -\frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \end{array} \right),\left( \begin{array}{cc} -\frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \end{array} \right), \left( \begin{array}{cc} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \end{array} \right),\left( \begin{array}{cc} -1 & 0 \\ 0 & 1 \\ \end{array} \right),\left( \begin{array}{cc} \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \end{array} \right).

    In fact, the Cayley graph for this group is as follows:

    cayley1

    For now, I will discuss how players can move within the action of 120 degree rotations. As in the previous posting, let the (x,y)-coordinates of player i be represented by (x^{i}, y^{i}), where i = 1,2,3,4,5. Then, under a 120 degree rotation, the player’s coordinates get shifted according to:

    \boxed{x^{i}_{t+1} = \frac{1}{2} \left(-x^{i}_{t} - \sqrt{3}y^{i}_{t}\right), \quad y^{i}_{t+1} = \frac{1}{2}\left(\sqrt{3}x^{i}_{t} - y^{i}_{t}\right)}

    This is a discrete dynamical system. In fact, it can be solved explicitly. Let x^i_{0}, y^{i}_{0} represent the initial coordinates of player i. Then, one solves the above discrete system to obtain:

    \boxed{x^i_t =\frac{1}{2} e^{\frac{1}{3} (-2) i \pi t} \left[\left(1+e^{\frac{4 i \pi t}{3}}\right) x^i_0+i \left(-1+e^{\frac{4 i \pi t}{3}}\right) y^i_0\right], \quad y^{i}_{t} =\frac{1}{2} e^{\frac{1}{3} (-2) i \pi t} \left[\left(1+e^{\frac{4 i \pi t}{3}}\right) y^i_0-i \left(-1+e^{\frac{4 i \pi t}{3}}\right) x^i_0\right]}

    Now, we can simulate this to see actually how players move within the triangle offense, forming equilateral triangles in every sequence:

    20160911_124208

    This is running in continuous time, that is, endlessly. In future postings, I will update this to include the other symmetries of the dihedral D_{3} group. However, the challenge is that this symmetry group is non-Abelian, so it will be interesting to implement pairs of consecutive symmetry operations in a simulation that would still result in invariant equilateral triangles.

    Hopefully, this post also shows why teams cannot really run “parts” of the triangle, as one player’s movement necessarily effects everyone else’s. This is something that Charley Rosen also mentioned in an article of his own.  

    The Possible Initial States of The Universe

    Most people when talking about cosmology typically talk about the universe in one context, that is, as a particular solution to the Einstein field equations. Part of my research in mathematical cosmology is to try to determine whether the present-day universe which we observe to be very close to spatially flat and homogeneous, and very close to isotropic could have emerged from a more general geometric state.

    What is often not discussed adequately is the fact that not only has our universe emerged from special initial conditions, but the fact that these special initial conditions also must include the geometry of the early universe, and the type of matter in the early universe. Below, I have attached a simulation that shows how the early universe can evolve to different possible states depending on the type of physical matter parametrized by an equation of state parameter \gamma . In particular, some examples are:

    • \gamma = 0: Vacuum energy
    • \gamma = 4/3: Radiation
    • \gamma = 2: Stiff Fluid

    Note: Click the image below to access the simulation!

    In these simulations, we present phase plots of solutions to the Einstein field equations for spatially homogeneous and isotropic flat, hyperbolic, and closed universe geometries. The different points are:

    1. dS: de Sitter universe – Inflationary epoch
    2. M: Milne universe
    3. F: spatially flat FLRW universe – our present-day universe
    4. E: Einstein static universe

    Note how by changing the value of \gamma , the dynamics lead to different possible future states. Dynamical systems people will recognize the problem at hand requires one to determine for which values of \gamma is F a saddle or stable node.