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 $$V(x) = x - x^* - x^* ln(\frac{x}{x^*}) \frac{k_1}{2}(T- T^*)^2 + \frac{k_2}{2}(U-U^*)^2$$$$\frac{dV}{dt} = \frac{\dot{x}}{x}( x - x^*) + other$$ 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 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
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