9.3 Stability of periodic orbits

Floquet theory is a powerful tool for analyzing periodic orbits in dynamical systems. It uses Floquet multipliers and the monodromy matrix to determine stability. This connects to Poincaré maps by providing a linear approximation of the map near periodic orbits.

Stable and unstable manifolds give a geometric picture of dynamics near periodic orbits. They separate regions with different behaviors and can intersect to create chaos. This ties into the broader theme of visualizing dynamics in phase space.

Floquet Theory

Analyzing Periodic Orbits with Floquet Multipliers and Monodromy Matrix

Top images from around the web for Analyzing Periodic Orbits with Floquet Multipliers and Monodromy Matrix

• 

  Stability Analysis of Periodic Solutions of Some Duffing’s Equations View original

  Is this image relevant?

• 

  Stability Analysis of Periodic Solutions of Some Duffing’s Equations View original

  Is this image relevant?

1 of 1

Top images from around the web for Analyzing Periodic Orbits with Floquet Multipliers and Monodromy Matrix

• 

  Stability Analysis of Periodic Solutions of Some Duffing’s Equations View original

  Is this image relevant?

• 

  Stability Analysis of Periodic Solutions of Some Duffing’s Equations View original

  Is this image relevant?

1 of 1

• Floquet theory provides a framework for analyzing the stability of periodic orbits in dynamical systems
• Floquet multipliers are eigenvalues of the monodromy matrix that characterize the stability of a periodic orbit
  • Multipliers with magnitude less than 1 indicate stable directions
  • Multipliers with magnitude greater than 1 indicate unstable directions
• The monodromy matrix represents the linear approximation of the Poincaré map for a periodic orbit
  • Obtained by integrating the variational equations along the periodic orbit over one period
  • Eigenvalues of the monodromy matrix are the Floquet multipliers

Quantifying Stability with Lyapunov Exponents

• Lyapunov exponents quantify the average exponential rates of divergence or convergence of nearby trajectories in the phase space
  • Positive Lyapunov exponents indicate chaos and instability
  • Negative Lyapunov exponents indicate stability and convergence
• For periodic orbits, Lyapunov exponents are related to the logarithms of the magnitudes of Floquet multipliers divided by the period
  • Provides a connection between Floquet theory and the long-term behavior of nearby trajectories

Manifolds

Stable and Unstable Manifolds

• Stable manifold of a periodic orbit consists of all points that approach the orbit as time goes to infinity
  • Trajectories on the stable manifold converge to the periodic orbit exponentially fast
  • Tangent to the eigenvectors corresponding to Floquet multipliers with magnitude less than 1
• Unstable manifold of a periodic orbit consists of all points that approach the orbit as time goes to negative infinity
  • Trajectories on the unstable manifold diverge from the periodic orbit exponentially fast
  • Tangent to the eigenvectors corresponding to Floquet multipliers with magnitude greater than 1

Significance of Manifolds in Understanding Dynamics

• Stable and unstable manifolds provide a geometric understanding of the dynamics near a periodic orbit
  • They separate the phase space into regions with different long-term behaviors
• Intersections of stable and unstable manifolds can lead to complex dynamics and chaos
  • Homoclinic and heteroclinic tangles (Smale horseshoe) indicate the presence of chaotic dynamics

Stability Types

Orbital Stability

• A periodic orbit is orbitally stable if nearby trajectories stay close to the orbit for all time
  • Small perturbations to the initial conditions result in trajectories that remain in a neighborhood of the periodic orbit
  • Characterized by Floquet multipliers with magnitude less than or equal to 1
• Orbital stability does not imply asymptotic stability
  • Nearby trajectories may not converge to the periodic orbit itself

Asymptotic Stability

• A periodic orbit is asymptotically stable if nearby trajectories converge to the orbit as time goes to infinity
  • Small perturbations to the initial conditions result in trajectories that approach the periodic orbit
  • Characterized by Floquet multipliers with magnitude strictly less than 1
• Asymptotic stability implies orbital stability
  • Nearby trajectories not only stay close to the periodic orbit but also converge to it over time
• Asymptotically stable periodic orbits are robust to small perturbations and can act as attractors in the phase space