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., d x d t = 0 \frac{dx}{dt} = 0 d t d x = 0 or d y d t = 0 \frac{dy}{dt} = 0 d t d y = 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