🔄Dynamical Systems Unit 4 – Phase Plane Analysis

Key Concepts and Definitions

• Phase plane a two-dimensional space where the axes represent the variables of a dynamical system
• Equilibrium points special points in the phase plane where the system remains at rest if it starts there
  • Also known as fixed points or steady states
• Stability describes the behavior of a system near an equilibrium point
  • Stable equilibrium point system returns to the point after a small perturbation
  • Unstable equilibrium point system moves away from the point after a small perturbation
• Nullclines curves in the phase plane where one of the derivatives is zero
• Vector field a graphical representation of the system's behavior at each point in the phase plane
• Limit cycles isolated closed trajectories in the phase plane that attract or repel nearby trajectories
• Bifurcation a qualitative change in the system's behavior as a parameter varies

Phase Plane Basics

• Phase plane analysis a graphical method for studying the behavior of two-dimensional dynamical systems
• Axes of the phase plane represent the state variables of the system (position and velocity)
• Trajectories curves in the phase plane that represent the evolution of the system over time
  • Also called solution curves or orbits
• Direction field a collection of arrows indicating the direction of the system's motion at each point
• Singular points points in the phase plane where both derivatives are simultaneously zero
• Separatrices special trajectories that separate regions of the phase plane with different behaviors
• Conservative systems systems where the total energy remains constant (frictionless pendulum)

Equilibrium Points and Stability

• Equilibrium points found by setting the derivatives of the system equal to zero and solving for the state variables
• Classify equilibrium points based on the eigenvalues of the Jacobian matrix evaluated at the point
  • Real, distinct eigenvalues node (stable if both negative, unstable if both positive)
  • Real, equal eigenvalues star node (stable if negative, unstable if positive)
  • Complex conjugate eigenvalues spiral or focus (stable if real part negative, unstable if real part positive)
  • Pure imaginary eigenvalues center (neutrally stable)
• Saddle points equilibrium points with one positive and one negative eigenvalue
  • Characterized by stable and unstable manifolds
• Lyapunov stability a system is stable if nearby trajectories remain close to the equilibrium point
• Asymptotic stability a system is asymptotically stable if nearby trajectories converge to the equilibrium point

Linearization Techniques

• Linearization approximating a nonlinear system by a linear one near an equilibrium point
• Jacobian matrix a matrix of partial derivatives evaluated at an equilibrium point
  • Eigenvalues and eigenvectors of the Jacobian matrix determine the local behavior near the equilibrium point
• Taylor series expansion a method for approximating a nonlinear function by a polynomial
  • First-order Taylor series expansion leads to the linearized system
• Hartman-Grobman theorem states that the behavior of a nonlinear system near a hyperbolic equilibrium point is qualitatively the same as that of its linearization
• Center manifold theorem a technique for reducing the dimension of a system by focusing on the center manifold near an equilibrium point
• Normal forms a simplified form of a nonlinear system obtained through coordinate transformations

Nullclines and Vector Fields

• Nullclines divide the phase plane into regions where the signs of the derivatives remain constant
  • $x$-nullcline set of points where $\frac{dx}{dt} = 0$
  • $y$-nullcline set of points where $\frac{dy}{dt} = 0$
• Intersection of nullclines determines the equilibrium points of the system
• Nullcline configuration provides insights into the global behavior of the system
• Vector field represents the direction and magnitude of the system's motion at each point
  • Arrows point in the direction of the system's motion
  • Length of arrows indicates the speed of the motion
• Stagnation points points in the vector field where the magnitude of the vector is zero
• Limit sets sets of points in the phase plane that trajectories approach as time goes to infinity (attractors) or negative infinity (repellers)

Limit Cycles and Periodic Orbits

• Limit cycles isolated closed trajectories in the phase plane
  • Stable limit cycle attracts nearby trajectories
  • Unstable limit cycle repels nearby trajectories
• Periodic orbits trajectories that repeat themselves after a fixed period
  • Correspond to limit cycles in the phase plane
• Poincaré-Bendixson theorem conditions for the existence of limit cycles in planar systems
• Hopf bifurcation a type of bifurcation that gives rise to limit cycles
  • Supercritical Hopf bifurcation creates a stable limit cycle
  • Subcritical Hopf bifurcation creates an unstable limit cycle
• Relaxation oscillations periodic orbits characterized by slow and fast motions (Van der Pol oscillator)
• Isoclines curves in the phase plane where the vector field has a constant slope

Applications in Real-World Systems

• Predator-prey models describe the interaction between two species (Lotka-Volterra equations)
  • Limit cycles represent sustained oscillations in population levels
• Epidemic models study the spread of infectious diseases (SIR model)
  • Equilibrium points correspond to disease-free and endemic states
• Neural networks model the behavior of interconnected neurons (Hodgkin-Huxley model)
  • Limit cycles represent repetitive firing patterns
• Biochemical reactions involve the interaction of chemical species (Brusselator model)
  • Periodic orbits correspond to sustained oscillations in concentrations
• Mechanical systems include pendulums, springs, and masses (Duffing oscillator)
  • Equilibrium points represent rest positions
  • Limit cycles correspond to self-sustained oscillations

Common Pitfalls and Tips

• Carefully choose the state variables to ensure the system is autonomous
• Normalize or rescale variables to simplify the equations and avoid numerical issues
• Pay attention to the direction of time when interpreting trajectories
• Use nullclines and vector fields to gain insights into the global behavior
• Check the stability of equilibrium points using the Jacobian matrix
• Be aware of the limitations of linearization techniques
  • Linearization is only valid near the equilibrium point
  • Nonlinear terms may dominate far from the equilibrium point
• Exploit symmetries and conserved quantities to simplify the analysis
• Use numerical simulations to complement analytical techniques
  • Verify analytical predictions
  • Explore the behavior for different parameter values