3.1 Phase portraits and equilibrium points

Phase portraits and equilibrium points are key tools for analyzing nonlinear systems. They give us a visual way to understand how a system's state changes over time, showing trajectories and critical points in a two-dimensional space.

By studying phase portraits, we can spot important behaviors like limit cycles and basins of attraction. Equilibrium points, where the system doesn't change, help us figure out long-term stability and how the system responds to small disturbances.

Phase Portraits for Nonlinear Systems

Graphical Representation and Vector Field

• Phase portraits graphically represent trajectories of a dynamical system in a two-dimensional phase plane with axes representing state variables
• Construct phase portraits by plotting the vector field determined by the system's differential equations describing the evolution of state variables over time
• Vector field represents the direction and magnitude of the rate of change of state variables at each point in the phase plane
• Observe patterns and shapes of trajectories in the phase portrait to analyze system behavior (limit cycles, separatrices, basins of attraction)

Nullclines and Equilibrium Points

• Nullclines are curves in the phase plane where one state variable's rate of change is zero
  • Horizontal nullcline: $\frac{dx}{dt} = 0$
  • Vertical nullcline: $\frac{dy}{dt} = 0$
• Intersections of nullclines represent equilibrium points where the system's state variables remain constant over time
• Analyze the system's long-term behavior and stability by examining nullclines and their intersections

Equilibrium Point Stability

Classification Based on Stability Properties

• Classify equilibrium points based on their stability properties describing the behavior of nearby trajectories
  • Stable equilibrium points attract nearby trajectories, causing convergence towards the point over time (asymptotically stable, marginally stable)
  • Unstable equilibrium points repel nearby trajectories, causing divergence away from the point over time
  • Saddle points attract trajectories along one direction and repel along another, creating a saddle-shaped pattern in the phase portrait
• Determine stability by analyzing eigenvalues of the system's Jacobian matrix evaluated at the equilibrium point

Determining Stability Using Eigenvalues

• Compute the Jacobian matrix $J(x^*, y^*)$ of the system evaluated at the equilibrium point $(x^*, y^*)$
• Calculate the eigenvalues $\lambda_1$ and $\lambda_2$ of the Jacobian matrix
• Classify stability based on eigenvalues:
  • Asymptotically stable: Both eigenvalues have negative real parts (Re$(\lambda_1) < 0$ and Re$(\lambda_2) < 0$)
  • Unstable: At least one eigenvalue has a positive real part (Re$(\lambda_1) > 0$ or Re$(\lambda_2) > 0$)
  • Marginally stable: All eigenvalues have non-positive real parts, and at least one has a zero real part (Re$(\lambda_1) \leq 0$ and Re$(\lambda_2) \leq 0$, with at least one equal to zero)

Trajectory Behavior Near Equilibrium Points

Types of Equilibrium Points

• Stable nodes: Nearby trajectories approach the point tangentially to the eigenvectors of the Jacobian matrix
• Unstable nodes: Nearby trajectories depart tangentially to the eigenvectors of the Jacobian matrix
• Stable foci: Nearby trajectories spiral towards the point, indicating damped oscillations
• Unstable foci: Nearby trajectories spiral away from the point, indicating growing oscillations
• Center points: Surrounded by closed orbits, indicating undamped oscillations

Separatrices and Basins of Attraction

• Separatrices are special trajectories that separate regions of the phase plane with different long-term behaviors
• Basins of attraction are regions in the phase plane where all trajectories converge to a specific equilibrium point or limit cycle
• Analyze separatrices and basins of attraction to understand the global behavior of the system and the influence of initial conditions on long-term dynamics

Equilibrium Point Stability Using Linearization

Linear Approximation and Jacobian Matrix

• Approximate the behavior of a nonlinear system near an equilibrium point using its linear approximation
• Compute the Jacobian matrix $J(x^*, y^*)$ of the system evaluated at the equilibrium point $(x^*, y^*)$
• The linear approximation is given by: $\dot{x} = J(x^*, y^*)(x - x^*)$

Stability Analysis Using Eigenvalues

• Determine the stability of the equilibrium point by analyzing the eigenvalues of the Jacobian matrix
  • Asymptotically stable: All eigenvalues have negative real parts
  • Unstable: At least one eigenvalue has a positive real part
  • Marginally stable: All eigenvalues have non-positive real parts, and at least one has a zero real part (further analysis using nonlinear techniques may be required)
• Hartman-Grobman theorem: The behavior of a nonlinear system near a hyperbolic equilibrium point is qualitatively similar to the behavior of its linear approximation

Limitations of Linearization

• Linearization is a local approximation and may not capture the global behavior of the system
• Nonlinear phenomena such as limit cycles, bifurcations, and chaos cannot be analyzed using linearization alone
• Combine linearization with other nonlinear analysis techniques (Lyapunov stability theory, center manifold theory) for a comprehensive understanding of the system's dynamics