4.3 Limit sets and attractors

Phase plane analysis helps us understand how dynamical systems behave over time. Limit sets and attractors are key concepts that reveal the long-term behavior of these systems, showing us where trajectories end up as time goes on.

Limit sets include omega-limit sets for future behavior and alpha-limit sets for past behavior. Attractors draw trajectories towards them, while repellers push them away. Understanding these concepts helps us predict system outcomes and identify important features.

Limit Sets

Defining Limit Sets

Top images from around the web for Defining Limit Sets

• 

  differential equations - Dynamical Systems- Plotting Phase Portrait - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Dynamical systems theory - Wikipedia View original

  Is this image relevant?

• 

  A simple test for asymptotic stability in some dynamical systems View original

  Is this image relevant?

• 

  differential equations - Dynamical Systems- Plotting Phase Portrait - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Dynamical systems theory - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Defining Limit Sets

• 

  differential equations - Dynamical Systems- Plotting Phase Portrait - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Dynamical systems theory - Wikipedia View original

  Is this image relevant?

• 

  A simple test for asymptotic stability in some dynamical systems View original

  Is this image relevant?

• 

  differential equations - Dynamical Systems- Plotting Phase Portrait - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Dynamical systems theory - Wikipedia View original

  Is this image relevant?

1 of 3

• Limit set represents the long-term behavior of a dynamical system
• Consists of all points that a trajectory approaches arbitrarily closely as time tends to infinity (positive or negative)
• Provides insight into the asymptotic behavior of a system (how it behaves as time goes to infinity)
• Can be used to identify attractors, repellers, and other important features of a dynamical system

Types of Limit Sets

• Omega-limit set contains points that a trajectory approaches as time tends to positive infinity
  • Represents the long-term future behavior of a system
  • Can be used to identify attractors (fixed points, limit cycles, or strange attractors)
• Alpha-limit set contains points that a trajectory approaches as time tends to negative infinity
  • Represents the long-term past behavior of a system
  • Can be used to identify repellers (unstable fixed points or limit cycles)

Properties of Limit Sets

• Limit sets are closed and invariant under the flow of the dynamical system
  • Closed means that if a point is in the limit set, all nearby points are also in the limit set
  • Invariant means that if a point is in the limit set, its entire trajectory (forward and backward in time) is also in the limit set
• Limit sets can be empty, finite, or infinite depending on the system
  • An empty limit set indicates that the trajectory escapes to infinity (unbounded system)
  • A finite limit set can consist of fixed points, limit cycles, or other simple structures
  • An infinite limit set can be a strange attractor or a chaotic set with complex structure

Attractors and Repellers

Defining Attractors and Repellers

• Attractor is a subset of the phase space that attracts nearby trajectories as time tends to infinity
  • Trajectories starting close to an attractor will converge to it over time
  • Examples include stable fixed points (sinks), stable limit cycles, and strange attractors
• Repeller is a subset of the phase space that repels nearby trajectories as time tends to infinity
  • Trajectories starting close to a repeller will diverge from it over time
  • Examples include unstable fixed points (sources) and unstable limit cycles

Basin of Attraction

• Basin of attraction is the set of all initial conditions that lead to trajectories converging to a specific attractor
  • Represents the region of influence of an attractor in the phase space
  • Trajectories starting within the basin of attraction will eventually approach the attractor
  • Basins of attraction can have complex shapes and boundaries, especially in nonlinear systems

Strange Attractors

• Strange attractor is an attractor with a fractal structure and chaotic dynamics
  • Exhibits sensitive dependence on initial conditions (nearby trajectories diverge exponentially)
  • Has a non-integer dimension (fractal dimension) and a complex geometric structure
  • Examples include the Lorenz attractor and the Rössler attractor
• Strange attractors are associated with chaotic behavior in nonlinear dynamical systems
  • Trajectories within a strange attractor are aperiodic and unpredictable, but confined to a bounded region
  • Chaotic systems have positive Lyapunov exponents, indicating exponential divergence of nearby trajectories

Equilibrium Points

Types of Equilibrium Points

• Stable node is an equilibrium point that attracts nearby trajectories from all directions
  • Eigenvalues of the Jacobian matrix at a stable node have negative real parts
  • Trajectories approach a stable node along the eigenvectors corresponding to the eigenvalues
  • Example: $\dot{x} = -x$, $\dot{y} = -y$ has a stable node at $(0, 0)$
• Unstable node is an equilibrium point that repels nearby trajectories in all directions
  • Eigenvalues of the Jacobian matrix at an unstable node have positive real parts
  • Trajectories diverge from an unstable node along the eigenvectors corresponding to the eigenvalues
  • Example: $\dot{x} = x$, $\dot{y} = y$ has an unstable node at $(0, 0)$
• Saddle point is an equilibrium point that attracts trajectories along some directions and repels along others
  • Eigenvalues of the Jacobian matrix at a saddle point have opposite signs (one positive, one negative)
  • Trajectories approach a saddle point along the stable eigenvector and diverge along the unstable eigenvector
  • Example: $\dot{x} = x$, $\dot{y} = -y$ has a saddle point at $(0, 0)$

Stability of Equilibrium Points

• Stability of an equilibrium point depends on the eigenvalues of the Jacobian matrix evaluated at that point
  • If all eigenvalues have negative real parts, the equilibrium point is stable (sink)
  • If all eigenvalues have positive real parts, the equilibrium point is unstable (source)
  • If the eigenvalues have opposite signs, the equilibrium point is a saddle point
• Linearization can be used to determine the local stability of an equilibrium point
  • Approximate the nonlinear system by its linear approximation (Jacobian matrix) near the equilibrium point
  • Analyze the eigenvalues of the Jacobian matrix to determine the stability
  • Linearization is valid only in a small neighborhood of the equilibrium point