Important Topics
- Routh Hurwitz Test
- Matrix of Variation/Jacobian of dx/dt (and dy/dt)
- Stability of solution
- Local Stability (Matrix of Variation) Eigenvalues
- Positive Definate : Unstable
- Negative Definate : Local AS
- Imaginary Eigenvalues: Spiral
- Global Stability (Appropriate Lyaponov Function)
- Test stability by Lyapunov Funcn or any other (V) has derivative negative definate
- Local Stability (Matrix of Variation) Eigenvalues
$$V(x) = x - x^* - x^* ln(\frac{x}{x^*}) \frac{k_1}{2}(T- T^*)^2 + \frac{k_2}{2}(U-U^*)^2$$
$$\frac{dV}{dt} = \frac{\dot{x}}{x}( x - x^*) + other$$
- Quick Finding of Eigenvalues (see Prerequisites)
- Complex Eigenvalues and Calculation of Spiral
- Linearization of Solution
- Logistic regression Model $$ \frac{dx}{dt} = rx(1 - \frac{x}{k})$$
- Persistance / Permanance of Solution
- Picard-Landlof Theorem - Existence of Solution
- Sylvester’s Criteria - b^2 - 4ac conditions
- Hamiltonian $$ H(x, t, u, \lambda) = g*{divident}(x, t, u) + \lambda f*{capital\ assets}(x, t, u) $$
- Pontrayagin’s Maximum Principle $$ \frac{d\lambda}{dt} = -\frac{dH}{dx}$$
- Bang-Bang and Singular Control (Control Theory)
- De Carte’s Rule of Sign
- Dulac Bendixson Criteria for periodicity of soln
- Bionic Equilibrium Conditions (for Optimal Harvesting)
- Hopf Bifurcation: The point where behavior of system stability changes. Opposite stability before and after critical value.
- Lebesgue Cycle Stability
- Basic Reproduction No.
- LimSup Method for showing boundedness
- Standard Comparison Theorem, Amax > Bmin, well-posedness …
Models
- Malthusian Growth Model $$ dx/dt = rx$$
- Logistic Growth Model (inter-specific interference)
- Resource-Consumer and similar models -
- Prey-Predator Model (or Resource Consumer)
- specialized prey-predator
- generalized prey-predator
- Competetive Model
- Cooperation Model
- Prey-Predator Model (or Resource Consumer)
- 3 Species Food Chain Model (Logistic Growth with interspecie interface)
- Opimal Harvesting (fish) Model - Max Sustainable Yield
- Migration of Fishes Model
- Pollution Toxicant Models-
- 2D Model
- 3D Model - Uptake of Conc (POST-Midsem)
- Susceptible-Infected and variant models
- SI Model
- SIS Model(with immunity)
- SIR Model(with complete cure forever)
- SEIR Model(Both Exposed and Recovery types)
Analysis of Solution
- Boundedness
- Positivity and Solution Space $\Omega$
- Persistance of Solution (Show Lower Bound)
- Periodicity or not (Dulac Bendixson Criteria)
- Equilibrium Points
- Local Stability Analysis
- Linearize solution and then find values OR…
- Use generalized matrix of variation and plug in values
- Global Stability Analysis
- Choose Lyapanov Function and terms based on Logistic Growth or not
- Differentiate and show Negative Definate
- Use Sylvester’s Criteria and compare terms using Routh-Hurwitz Criteria
- Routh Array to show stability
Other Analysis Techniques
- Rate $\dot{r}$ for growth and $\dot{\theta}$ for clockwise/anticlockwise in spirals
- Lebesgue Cycle stability in spirals
- Critical points in Hopf Bifurcation and stability chart
- Basic Reproduction Number Calculation
Sample Model to check equilibrium points
graph TD
A[Formulate the Rate Diffn Equations] --> AB
AB[ Find Omega. Check if bounded with limsup and show positive also. Check Persistance/Periodicity] --> B
B[Find equilibrium points where rate = 0] --> C
C[Local Stability. Get matrix of variation/Jacobian at these points] --> D
C --> E[Get Char Eqn with Eigenvalue]
E --> F[Use Routh Hurwitz to get roots' sign]
F --> G[From sign of eigenvalues determine stability of Local Solution]
D[Get Eigenvalues of the matrix of variation] --> G
G --> H
H[Global Stability. Use Lyapunov function variants Derivative] --> I[Check if negative definate by adding/subtracting]
Harvesting Model Sample Flow
Steps for Solving
- Formulate the Rate Diffn Equations
- Find the equilibrium points and conditions
- Plug x* value in harvesting rate equation qEx
- Get max E by finding minima at x* and put the value into harvesting rate
For optimal harvesting policy,
- Find hamiltonian with - f as net revenue in continuous time stream - g as rate of change of assets $$ H = e^{-\delta t}(pqx-c)E + \lambda_1 \frac{dx}{dt} +\lambda_2 \frac{dy}{dt}$$
- Find minima of H wrt. E .Get switching func and equate to 0 for singular control
- Apply Pontrayagin’s Max Principle Condition and use with step 6.
- Find discount value and check notes of these points
Bionic Equilibrium is value of E at $$\dot{x} = \dot{y} = 0$$
SIR Model
Steps for Solving,
- Show Bounded by taking sum of population(N) and finding LimSupN(t) will be const
- Take S(t) and with Comparison Theorem, show >=0
- Use Dulac Bendixson with H = 1/SI, and show no sign change so not periodic
- Find Equilibrium Points (approx E*) with reproduction no. ‘R’
- Matrix of Variation with ‘R’ cases
- Lyapunov Function and one term will not allow sylvester’s criteria
- Choose C = S* to get, $$ \dot{V} = -\beta I (S - S^*) -\beta S^* (I-I^*)( S - S^* ) < 0 $$
Other Useful Prerequisites
Green’s Thm
Stoke’s Thm
Gauss Divergent Thm
Finding Eigenvalues fast $$ \lambda^2 - Tr(A)\lambda + Det(A) $$ Trace is sum of diagonal entries
Routh Hurwitz Criteria See Routh Hurwitz Test - Conditions - All positive coeff always - All minors must be positive - 2nd Degree: no more conditions - 3rd Degree a1*a2 - a3*a0 > 0 (for 3 degree see below)
$$ Routh\ Array = \begin{bmatrix} a_{N} & - a_{N-2} \ a_{N-1} & a_{N-3} \ \end{bmatrix} $$
Another view to understand Routh Hurwitz Graphical
- Background
- If one sign different => one is positive. Hence unstable.
- If all negative => same as all positive coeff
- Background