🪝Ordinary Differential Equations Unit 10 – Analyzing Differential Equations Qualitatively

Key Concepts and Definitions

• Ordinary differential equations (ODEs) mathematical equations involving functions and their derivatives
• Qualitative analysis focuses on the behavior of solutions without explicitly solving the equations
• Phase space abstract space in which all possible states of a system are represented (each point corresponds to a unique state)
• Equilibrium points special points in the phase space where the system remains unchanged over time
  • Also known as fixed points or steady states
• Stability refers to the behavior of solutions near an equilibrium point
  • Stable equilibrium points solutions starting nearby converge to the point
  • Unstable equilibrium points solutions starting nearby diverge from the point
• Bifurcations qualitative changes in the behavior of a system as a parameter varies
• Nullclines curves in the phase plane where one of the derivatives is zero

Types of Differential Equations Covered

• First-order ODEs involve only the first derivative of the dependent variable
  • Examples: $\frac{dy}{dt} = f(t, y)$, $\frac{dx}{dt} = ax + by$
• Second-order ODEs involve the second derivative of the dependent variable
  • Example: $\frac{d^2y}{dt^2} + a\frac{dy}{dt} + by = 0$
• Autonomous ODEs do not explicitly depend on the independent variable (usually time)
  • Example: $\frac{dx}{dt} = x(1-x)$
• Non-autonomous ODEs explicitly depend on the independent variable
  • Example: $\frac{dy}{dt} = \sin(t) - y$
• Linear ODEs have linear combinations of the dependent variable and its derivatives
  • Example: $\frac{dy}{dt} + 2y = e^t$
• Nonlinear ODEs involve nonlinear combinations of the dependent variable and its derivatives
  • Example: $\frac{dx}{dt} = x^2 - 4x$

Qualitative Analysis Techniques

• Direction fields (slope fields) graphical representation of the solution curves' slopes at various points in the phase plane
  • Helps visualize the general behavior of solutions without explicitly solving the ODE
• Isoclines curves in the direction field along which the solution curves have the same slope
  • Useful for sketching solution curves and identifying equilibrium points
• Linearization approximating a nonlinear system near an equilibrium point by a linear system
  • Helps determine the local stability of equilibrium points
• Nullcline analysis finding curves where one of the derivatives is zero and studying their intersections
  • Intersections of nullclines are equilibrium points
• Lyapunov functions special functions used to prove the stability of equilibrium points
  • If a Lyapunov function exists and satisfies certain conditions, the equilibrium point is stable
• Poincaré-Bendixson theorem states that a bounded solution of a 2D autonomous system must approach an equilibrium point, a limit cycle, or a union of equilibrium points and trajectories connecting them

Phase Plane Analysis

• Phase plane (phase space) abstract space in which all possible states of a system are represented
  • Each axis represents one of the dependent variables or its derivative
• Trajectories (solution curves) paths in the phase plane that represent the evolution of the system over time
  • Tangent vectors to the trajectories indicate the direction of motion
• Vector field (direction field) assigns a vector to each point in the phase plane, indicating the direction and magnitude of change
• Equilibrium points (fixed points, steady states) points in the phase plane where all derivatives are zero
  • Represented by the intersections of nullclines
• Limit cycles isolated closed trajectories in the phase plane
  • Solutions nearby a stable limit cycle approach it over time
• Separatrices special trajectories that separate regions of the phase plane with different qualitative behaviors

Equilibrium Points and Stability

• Equilibrium points (critical points, fixed points) points in the phase space where the system remains unchanged over time
  • Mathematically, all derivatives are zero at equilibrium points
• Classification of equilibrium points depends on the eigenvalues of the linearized system
  • Sink (stable node) all eigenvalues have negative real parts, nearby solutions converge to the point
  • Source (unstable node) all eigenvalues have positive real parts, nearby solutions diverge from the point
  • Saddle point some eigenvalues have positive real parts, others have negative real parts, nearby solutions converge along some directions and diverge along others
  • Center purely imaginary eigenvalues, nearby solutions follow closed orbits around the point
• Stability determines the long-term behavior of solutions near an equilibrium point
  • Asymptotic stability solutions starting nearby converge to the equilibrium point as time approaches infinity
  • Instability solutions starting nearby diverge from the equilibrium point
• Lyapunov stability solutions starting close to the equilibrium point remain close for all future time
  • Does not necessarily imply asymptotic stability

Bifurcations and Parameter Changes

• Bifurcations qualitative changes in the behavior of a system as a parameter varies
  • Examples: changes in the number or stability of equilibrium points, appearance or disappearance of limit cycles
• Bifurcation points critical parameter values at which bifurcations occur
• Saddle-node (fold) bifurcation two equilibrium points collide and annihilate each other as the parameter varies
• Transcritical bifurcation two equilibrium points exchange their stability as they pass through each other
• Pitchfork bifurcation one equilibrium point splits into three (or vice versa) as the parameter varies
  • Supercritical pitchfork bifurcation stable equilibrium point becomes unstable, and two new stable equilibrium points appear
  • Subcritical pitchfork bifurcation unstable equilibrium point becomes stable, and two unstable equilibrium points disappear
• Hopf bifurcation an equilibrium point changes stability, and a limit cycle appears or disappears
  • Supercritical Hopf bifurcation stable equilibrium point becomes unstable, and a stable limit cycle appears
  • Subcritical Hopf bifurcation unstable equilibrium point becomes stable, and an unstable limit cycle disappears

Applications and Real-World Examples

• Population dynamics models the growth and interactions of populations over time
  • Logistic equation $\frac{dP}{dt} = rP(1-\frac{P}{K})$ describes the growth of a population with limited resources
  • Lotka-Volterra equations $\frac{dx}{dt} = \alpha x - \beta xy, \frac{dy}{dt} = \delta xy - \gamma y$ model predator-prey interactions
• Epidemiology studies the spread of infectious diseases in a population
  • SIR model $\frac{dS}{dt} = -\beta SI, \frac{dI}{dt} = \beta SI - \gamma I, \frac{dR}{dt} = \gamma I$ describes the dynamics of susceptible, infected, and recovered individuals
• Chemical kinetics analyzes the rates of chemical reactions
  • Example: $\frac{d[A]}{dt} = -k[A], \frac{d[B]}{dt} = k[A]$ models a first-order reaction $A \rightarrow B$
• Mechanical systems ODEs can describe the motion of objects subject to forces and constraints
  • Example: $\frac{d^2x}{dt^2} + \frac{k}{m}x = 0$ represents a simple harmonic oscillator (mass-spring system)
• Electrical circuits ODEs model the behavior of currents and voltages in electrical networks
  • Example: $L\frac{dI}{dt} + RI = V(t)$ describes a series RLC circuit with a time-varying voltage source

Common Pitfalls and Tips

• Ensure that the direction field or vector field is consistent with the given ODE
  • Vectors should be tangent to the solution curves at each point
• Be careful when sketching solution curves in the phase plane
  • Solution curves cannot cross each other (except at equilibrium points)
  • Solution curves should follow the direction indicated by the vector field
• Remember that linearization is only valid near the equilibrium point
  • The behavior of the nonlinear system may differ from the linearized system far from the equilibrium point
• Check the stability of equilibrium points using the eigenvalues of the linearized system
  • Negative real parts imply stability, positive real parts imply instability
• Consider the possibility of limit cycles and other attractors in nonlinear systems
  • Poincaré-Bendixson theorem can help identify the presence of limit cycles in 2D autonomous systems
• Be aware of the limitations of qualitative analysis
  • Some aspects of the system's behavior may not be captured by the qualitative techniques
  • Quantitative methods (numerical simulations, explicit solutions) can provide additional insights
• Pay attention to the units and physical meaning of variables and parameters in applications
  • Ensure that the ODEs and their solutions are dimensionally consistent and interpretable in the context of the problem