10.1 Equilibrium Points and Phase Portraits

Equilibrium points are crucial in understanding differential equations. They represent constant solutions where change stops, and come in various types like stable, unstable, and saddle points. Knowing these helps predict how solutions behave over time.

Phase portraits visually show a system's behavior in the phase plane. They include trajectories, vector fields, and nullclines, revealing key features like equilibrium points, periodic solutions, and long-term trends. This graphical approach offers valuable insights into complex systems.

Equilibrium Points

Types of Equilibrium Points

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• Equilibrium point represents a constant solution to a differential equation where the rate of change is zero
• Stable equilibrium attracts nearby solutions causing them to converge towards it over time (attractor)
• Unstable equilibrium repels nearby solutions causing them to diverge away from it over time (repeller)
• Saddle point attracts solutions along one direction while repelling them along another direction
  • Results in a hyperbolic trajectory near the equilibrium point
• Node is a stable or unstable equilibrium point where all trajectories approach or leave the point along a straight line (eigenvalues are real and have the same sign)
• Focus is a stable or unstable equilibrium point where trajectories spiral towards or away from the point (eigenvalues are complex with nonzero real part)
• Center is a neutrally stable equilibrium point surrounded by closed orbits (eigenvalues are purely imaginary)
  • Small perturbations result in a different closed orbit around the center

Classifying Equilibrium Points

• Equilibrium points can be classified based on the eigenvalues of the Jacobian matrix evaluated at the point
  • Eigenvalues determine the stability and behavior of solutions near the equilibrium
• A stable node has negative real eigenvalues (sink)
• An unstable node has positive real eigenvalues (source)
• A saddle point has real eigenvalues with opposite signs
• A center has purely imaginary eigenvalues
  • Indicates a conservative system with no damping
• A stable focus has complex eigenvalues with negative real part
• An unstable focus has complex eigenvalues with positive real part
  • Imaginary part causes spiraling behavior

Phase Portraits

Elements of a Phase Portrait

• Phase portrait is a graphical representation of the trajectories of a dynamical system in the phase plane
  • Provides a qualitative view of the system's behavior for various initial conditions
• Trajectory represents the path or solution curve that a system follows over time for a given initial condition
  • Tangent vectors along a trajectory indicate the direction and magnitude of change
• Vector field shows the direction and relative magnitude of the system's rate of change at each point in the phase plane
  • Represented by arrows or line segments
• Nullcline is a curve in the phase plane along which one of the differential equation's components is zero
  • Equilibrium points occur where nullclines intersect

Interpreting Phase Portraits

• Phase portraits reveal important features of a system's behavior:
  • Location and type of equilibrium points
  • Basins of attraction for stable equilibria
  • Separatrices dividing regions with different long-term behavior
• Closed orbits indicate periodic solutions (limit cycles)
  • Can be stable, unstable, or semi-stable
• Homoclinic and heteroclinic orbits connect saddle points to itself or other saddles
• Studying the phase portrait near an equilibrium point provides insight into its local stability
  • Linearization can be used to approximate the nonlinear system near the equilibrium
• Global behavior is inferred by examining the overall structure of the phase portrait
  • Identifies attractors, repellers, and other invariant sets