# MATH3063 Nonlinear ODEs with Applications

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Flashcard maker: Fergus O' Sullivan
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## Decks in this class (12)

1 - First order equations
First-order equations: Mathematical models–exponential and logistic growth; definition of linear and nonlinear, autonomous and non-autonomous. Phase portraits, equilibria, stability (using phase portraits), linear stability, linearisation for single first order equations.
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2 - Bifurcations
Bifurcation in single, first-order equations, including the exponential and logistic models. Harvesting, both constant rate and constant effort. Population catastrophes. Bifurcation diagrams. Hysteresis.
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3 - Introduction to predator-prey models
Introduction to predator-prey models, specifically the Lotka-Volterra model. Nonlinear models cannot be solved explicitly, hence the need for new mathematical tools. Introduction of the phase plane; nullclines, flows, sketching solutions. Phase plane of the Lotka-Volterra equations motivates the need for more information.
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4 - Nonlinear systems and linearisation
The creator of this deck did not yet add a description for what is included in this deck.
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5 - Phase portraits and linear stability
Phase portraits and linear stability analysis of a variety of nonlinear systems (lots of examples). Existence and uniqueness of solutions. Different types of stability
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6 - Lotka-Volterra equations
Lotka-Volterra equations using linear analysis and phase planes. Harvesting the Lotka-Volterra equations. Structural instability. Other predator-prey systems. Models for ecological competition, mutualism.
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7 - Models for the spread of disease
Models for the spread of disease. Definition of an epidemic. Basic SIR model. Critical population sizes. Vaccination effects. What happens as t → ∞. SIS and SIRS models, crisscross infections and STDs.
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8 - Lyapunov stability
Lyapunov stability. Finding and using Lyapunov functions. Sketch of Lyapunov theorems.
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9 - First integrals, Hamiltonian systems and gradient systems
First integrals, Hamiltonian systems and gradient systems. Definition of first integral. The Lotka-Volterra equations as an example of a Hamiltonian system. Conservative systems. nonlinear pendulum, Duffing equation, the Van der Pol oscillator.
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10 - Limit cycles
Limit cycles: definition, stability analysis, phase protraits. Biological examples (mainly computational).
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11 - Bifurcation in systems of two first order ODEs
Bifurcation in systems of two first order ODEs. Statement of the Hopf bifurcation theorem. Creation of limit cycles. The Brusselator model. Predator-prey and epidemiological models with Hopf bifurcations.
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12 - Fitzhugh-Nagumo equations
Fitzhugh-Nagumo equations, relaxation oscillations and excitable media
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