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Practicum 1 (February 14)
introduction: ODE -- definition, concept of solution
ODE type: with separable variables
demo: x'=3t², x'=5x
class: x'=x²+1, x'=e^x(t+1) , x'=(x^2-x)/t
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Practicum 2 (February 21)
demo: x'=2√|x|e^(-t)
ODE type: 1st order linear: integrating factor
demo: x'+(cos t)x = (1+t)e^(-sin t)
class:
x'-x=te^t
x'-x/t = t^2e^t
x-x/t^2 = 1/t^3
x'+2x=cos t
ODE type: Bernoulli (reduction to linear)
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Practicum 3 (February 28)
ODE type: (Bernoulli) and homogeneous
demo:
x' -4x/t = t√x
t²x'+tx = t²+x²
class:
1. x'=exp(x/t)+x/t
2. x'-2tx = 2t³x²
3. x' = (x-t)/(x+t)
4. x' - 9t²x = (t⁵+t²)\root{3}\of{x²}
5. t²x' = x² + 2tx
6. 2txx' + t²-x² = 0
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Practicum 4 (March 7)
Elementary qualitative analysis for x'=f(t,x)
demo: x'=e^x(t+1) , x'=2√|x|e^(-t)
class:
1. x'=t²(x+1)
2. x'=(x-1)/(t-1)
3. x'=t(x+1)
4. x'=x/t+t²
5. x'=2tx-2
Remark on symmetry
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Practicum 5 (March 14)
demo: x'=x²+t²-1
EQA for autonomous systems in R²
demo: x'=x²-2x+y , y'=y(1-x)
class:
1. x'=x(1-x)-xy , y'=-2y+xy
2. x'=x(1-x/2-y) , y'=y(2-2x-y)
3. x'=x(2-2x-y) , y'=y(1-x/2=y)
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Practicum 6 (March 21)
[More like a lecture than a practicum.]
1. ODE classification (general system, autonomous, ...)
2. Existence and uniqueness (Peano and Picard).
Examples.
3. Stability and asymptotic stability.
Definitions. Theorem on linearized stability.
Example:
x'=x(2-2x-y) , y'=y(1-x/2=y)
E₁=(1,1) is asymptotically stable
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Practicum 7 (March 28)
In class: investigate stability of equilibria
1. x'=2x+y-1 , y'=sin(x)-y²+1
2. x'=ln(5-2x-2y), y'=exp(xy)-1
3. x'=(1-x-y/(x+1/4))x, y'=(1-4y/3x)y
4. x''+ αx' + sin(x) = 0
5. x'=1-x²-y², y'=z²-x-y, z'=z²-1
Theorem on stable/unstable direction.
Examples:
x'=(1-x/2-y)x, y'=(2-2x-y)y where E=(2/3,2/3)
x'+sin(x)=0 where E=(0,0) and (0,π)
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Practicum 8 (April 4)
first integral
demo: x'' + sin(x) (pendulum continued)
x' = (x-t)/(x+t)
x'=x²y , y' = xy²
In class:
1. x'=xy³ , y'=xy+xy²
2. x'=x+y, y'=x²-y²
3. x''+x = 0 (later as demo)
4. x''+x^2-x=0
5. x'=xy , y'=xz , z'=yz
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Practicum 9 (April 11)
Linearization theorems - remarks on the proof.
Verification of 1st integral (Theorem P.7)
Examples: x'=x²y , y' = xy², SIR model
In class: as last time, including Lotka-Volterra system
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Practicum 10 (April 25)
topic: linear homogeneous ODE with constant coefficients
(L-1) system of n equations
(L-2) 1 equation of n-th order
HOWTO solve (L-2): characteristic polynomial,
fundamental system (via exponentials/polynomials)
demo: x"+2x'-3x=0, x"+x=0, x"+2x'+x=0
In class:
1. x""-3x"+2x=0
2. x'''+3x''+3x'+x=0
3. x""+18x"+81x=0
4. x'=10x-6y, y'=18x-11y
5. x'=y, y'=z, z'=y
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Practicum 11 (May 2)
demo: x'=2x+y, y'=x-2y
theory: Theorem P-8, matrix exponential
In class:
1. x'=-12x-16y, y'=9x+12y (nilpotent)
2. x'=-y, y'=x (±i)
3. x'=y+2z, y'=3z, z'=0 (nilpotent)
4. A³+A²=0 (spectrum? exponential?)
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Practicum 12 (May 9)
Matrix exponential - some properties
Problem: classification of 2x2 linear systems
case (1): λ₁<λ₂<0
case (2): 0<λ₂<λ₁
topic: numerical software & numerical methods
demo: "plotdf" command in Maxima (https://maxima.sourceforge.io/)
example continued: x'=x²-2x+y , y'=y(1-x)
... first integral: V = x²(1+y)²+y²+2y³/3
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Practicum 13 (May 16)
Example finished (quickly).
Topic: numerical methods - simple examples
of first and second order difference method
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Lecture 1 (April 4)
game theory - an introduction.
keywords: player, strategy, payoff
key concept: Nash equilibrium (N.e.)
Examples:
rock-scissors-paper (RSP), passenger-inspector
prisoner's dilemma (PD)
generalizations: mixed strategies, repeated games
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Lecture 2 (April 11)
Definitions: normal form 2-player game.
Pure and mixed strategies.
Payoff matrices. Best reply.
Lemma 7.1. Characterisation of best replies as
a convex combination of pure best replies.
Definition: Nash equilibrium (N.e.)
Theorem 7.1. Existence of Nash equilibrium.
Example: computing N.e. for "passenger-inspector" game.
Remarks. 1) generalization: N-player (normal form) game
2) extended form game (random moves, incomplete information)
Zermelo chess theorem.
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Lecture 3 (April 25)
Simplification: only symmetric games from now on.
Definition: evolutionary stable strategy (ESS).
Lemma 7.2. Characterisation of (ESS) - relation to N.e.
Replicator dynamics: axiomatic approach.
Useful observation: game normalization.
Example: non-symmetric RSP game (without proofs mostly)
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Lecture 4 (May 2)
RD: elementary derivation
Theorem 7.2. Well-posedness of (RD); invariance of support.
Theorem 7.3. Equilibria of (RD) vs. N.e.
Theorem 7.4. ESS => asymptotically stable for (RD)
Example (continuation): non-symmetric RSP game
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Lecture 5 (May 9)
Theorem 7.4. -- very brief comment on the proof
Question: Does (RD) increase π(x) ? (the average gain)?
Example: HD, extension HDBR
Theorem 7.5. [Fisher's fundamental theorem of natural selection.]
RD - critique & improvement - random "mutations" (not finished)
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Lecture 6 (May 16)
RD - with mutations: simplified with uniform mutations
Example: PD repeated PD with mutations
Lemma 7.3. more on the stationary points of RD
Theorem 7.6. negative criterion for periodic orbit and stat. points
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