3.3 Phase portraits for linear systems

Linear systems are the foundation of dynamical systems theory. Phase portraits help visualize the behavior of these systems, showing how variables change over time. They reveal equilibrium points, stability, and trajectories, giving us a clear picture of the system's dynamics.

Understanding phase portraits is crucial for analyzing more complex nonlinear systems. By studying the patterns and structures in these portraits, we can predict long-term behavior and stability of various systems in physics, biology, and engineering.

Equilibrium Points and Stability

Types of Equilibrium Points

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• Equilibrium points are steady-state solutions where the system remains unchanged over time
• Stable node is an equilibrium point that attracts nearby trajectories, causing them to converge towards it (sink)
• Unstable node is an equilibrium point that repels nearby trajectories, causing them to diverge away from it (source)
• Saddle point is an equilibrium point that attracts trajectories along one direction and repels them along another, creating a saddle-shaped pattern
• Center is an equilibrium point surrounded by closed orbits, where trajectories neither converge nor diverge but instead form concentric circles or ellipses

Determining Stability

• Stability of equilibrium points can be determined by analyzing the eigenvalues of the Jacobian matrix evaluated at the equilibrium point
• For a stable node, all eigenvalues have negative real parts, indicating exponential decay of perturbations
• For an unstable node, all eigenvalues have positive real parts, indicating exponential growth of perturbations
• For a saddle point, eigenvalues have real parts with opposite signs, leading to attraction along one direction and repulsion along another
• For a center, eigenvalues are purely imaginary, resulting in oscillatory behavior without convergence or divergence

Trajectories and Nullclines

Trajectories and Phase Portraits

• Trajectories represent the solutions or paths followed by the system in the phase space over time
• Phase portraits visualize the qualitative behavior of a system by plotting multiple trajectories in the phase space
• Trajectories in a phase portrait provide insights into the long-term behavior and stability of the system
• The direction and shape of trajectories in a phase portrait depend on the underlying differential equations and initial conditions

Nullclines and Special Points

• Nullclines are curves in the phase space where one of the variables' rates of change is zero (e.g., $\frac{dx}{dt} = 0$ or $\frac{dy}{dt} = 0$)
• Intersection points of nullclines are equilibrium points, as both variables' rates of change are simultaneously zero
• Spiral point is an equilibrium point around which trajectories spiral inward (stable spiral) or outward (unstable spiral)
• Limit cycle is an isolated closed trajectory in the phase space that attracts or repels nearby trajectories, representing a periodic solution

Analyzing Phase Portraits

• Nullclines divide the phase space into regions where the signs of the variables' rates of change differ, determining the direction of trajectories
• Arranging nullclines and identifying equilibrium points help construct the overall phase portrait
• Studying the behavior of trajectories near equilibrium points and nullclines reveals the system's qualitative dynamics
• Examples of systems exhibiting limit cycles include the Van der Pol oscillator and the Fitzhugh-Nagumo model in neuroscience