1. SIAM Conference on Applications of Dynamical Systems Snowbird, Utah 17-21 May 2009 Distinguished trajectories in time dependent geophysical flows Ana M. Mancho & JA Jiménez Madrid 1 Instituto de Ciencias Matemáticas, CSIC 1 Now at Cambridge University

2. Autonomous Dynamical Systems Fixed points are a keystone to describe solutions in autonomous DS. They are on the basis of the geometrical description of the flow. Stable and unstable manifolds of hyperbolic fixed points organize the flow in the phase space. Manifolds are made of trajectories that asymptotically (in plus or minus infinity time) approach the fixed points. They are barriers to transport that particles cannot cross Integrable Duffing Fixed points Centers Saddle points Autonomous Dynamical Systems x 1 = x 2 x 2 = x 1 -x 1 3 x= (0, 0)

3. Periodic Dynamical Systems Periodic orbits are an essential tool for the geometrical description of periodic flows. Periodic orbits are fixed points of the Poincaré map This solution is obtained perturbatively from the fixed point solution x =(0,0). x 1 = x 2 x 2 = x 1 -x 1 3 +  sin t x periodic = -  /2 (sin t, cos t) + O(  3 ).. Periodic Dynamical Systems

4. A piece of unstable manifoldA piece of stable manifold Stable and unstable manifolds of hyperbolic periodic orbits organize the flow in the phase space. Periodic Dynamical Systems

5. Generalized fixed points There exist examples of trajectories in time dependent linear systems that generalize the concept of fixed point. 1D example All trajectories asymptotically tend to the solution x p in plus infinity time. This fits into the idea of generalized fixed point. We attempt to provide a unique general definition that contains solutions such as fixed points, periodic orbits and this particular solution. Generalized fixed points

6. Distinguished trajectories Distinguished trajectory ‘Move less’ is satisfied by the minimum of a function M which measures the length of the displacement of a trajectory forwards and backwards in time on the phase space. The distinguished trajectory is that which ‘moves less’.

7. Distinguished trajectories Distinguished trajectory At time t*=0, different initial conditions x* have different displacements, but one condition has a local minimum for M. Different  values provide different initial condition x* for the minimum. Definition. (-Distinguished trajectory). A trajectory  ( t ) of the system is -distinguished at time t* if there exists an open set B around  (t*) where the defined function M(x*) t*,  has a minimum and min( M(x*) t*,  )= M(  (t*)) t*, 

8. Distinguished trajectory Different -values provide different initial condition x* for the minimum. However the minimum of M converges for increasing  to the limit coordinates The value of the limit coordinate x*=-1 corresponds to the initial condition at time t=0, for the particular solution x p =t-1 We have been able to determine the coordinates x* of the searched solution at a time t* as it satisfies the property of being a local minima of M for large enough  Distinguished trajectories

9. Distinguished trajectory A trajectory  (t) is said to be Distinguished with accuracy in a time interval [ t 0, t N ] if there exist a continuous path of limit coordinates ( t l,x l ), such that Where the distance ||. || is defined as follows: Distinguished trajectories

10. Applications Periodic Duffing Equation Hyperbolic Periodic Trajectory Structure of M for increasing 

11. Applications Periodic Duffing Equation Limit coordinates Computational DHT

12. Applications Periodic Duffing Equation Non Hyperbolic Periodic Trajectory Structure of M for increasing 

13. Applications Periodic Duffing Equation Limit coordinates Computational DHT

14. Applications A 3D extension of the Duffing Equation Computational DHT

15. Applications Geophysical flows: finite time data sets Numerical data from a QG Model

16. Geophysical flows: finite time data sets In Mancho, A. M., Small, D., Wiggins, S., 2004. Nonlinear Process. Geophys. 11, 17 33, this DHT is reported for the finite time data set for a time interval in the Northern gyre The proposed method could not be applied beyond a certain day What happens to the distinguished trajectory next? Where it goes? Applications

17. Geophysical flows: finite time data sets Path of limit coordinates and trajectories. Beyond day 300 they do not coincide. Distinguished trajectories stop being distinguished! Applications Computational DHT

18. Geophysical flows: satellite velocity data Applications (Image courtesy of Carolina Mendoza). Function M at the Gulf stream on the 20th November 2002. Locates hyperbolic and non hyperbolic points at a glance

19. Conclusions A definition of Distinguished trajectory has been proposed which recovers fixed points of autonomous systems, periodic orbits of periodically forced systems This definition is useful to compute DT which are not hyperbolic The definition is proposed for dynamical systems of general dimension n When applied to highly chaotic and aperiodic vector fields the definition is useful to compute trajectories that held the property of being distinguished in a certain time interval. These trajectories are important for the geometrical description of the flow. More details JAJ Madrid, AM Mancho. Distinguished trajectories in time dependent vector fields. http://arXiv.org/abs/0806.1155. Chaos 19, 013111 (2009) http://arXiv.org/abs/0806.1155 Conclusions