A method for visualization of invariant sets of dynamical systems based on the ergodic partition

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1. What contributions have the authors mentioned in the paper "A method for visualization of invariant sets of dynamical systems based on the ergodic partition" ?

2. What have the authors stated for future works in "A method for visualization of invariant sets of dynamical systems based on the ergodic partition" ?

3. What is the common method for the visualization of phase space structure in four-dimensional dynamical?

4. What is the meaning of ergodic partitions?

Accepted Answer

The authors provide an algorithm for visualization of invariant sets of dynamical systems with a smooth invariant measure.

Accepted Answer

Previously used methods for visualization were the following: • Plotting of orbits, which can be used effectively only in two-dimensional systems—already in three dimensions the authors are forced to project to two-dimensional planes of the computer screen and paper.. The authors have shown that ergodic partition can be constructed out of sets on which the time-averages of a dense, countable set of functions in the space of all continuous functions are constant.. In the numerical implementation of the algorithm the authors can take only finite-time averages.. In some sense, their algorithm can be considered as a finite-time measurement of the system in which the authors are interested in the average properties.

Accepted Answer

A common method for the visualization of phase space structure in four-dimensional dynamical systems involves the color-coding of invariant surfaces ~in particular, of stable and unstable manifolds of certain phase-space objects!

Accepted Answer

The stationary partition z is called ergodic if, for almost every (with respect to m) element C of z, there is an invariant measure mC on C such that the restriction of T to C , denoted TC is an ergodic automorphism on C , with respect to some probability measure mC on C , and, for every f PL1(A),E A f dm5E A F E C f uCdmCGdm , ~1!where f uC denotes the restriction of f to the ergodic component C .The notion of the ergodic partition of automorphisms of Lebesgue spaces has its origin in the works of von Neumann ~1932!; Halmos ~1941, 1949!; and Rokhlin ~1949!.

Accepted Answer

There are many issues that the authors only touched upon here, like the effect of dimensionality and the variation of functions on the convergence of the time averages, the choice of functions, the numerical implementation of the product operation, etc.

Accepted Answer

The method of plotting exit times can be applied for dynamical systems of arbitrary dimension, as the phase-space is color coded with a particular color assigned depending on the time it takes to get out of the prescribed, bounded set in the phase space ~see Easton et al., 1993!.

Accepted Answer

The numerical algorithm consists of computing the time averages of a chosen set of functions and partitioning the phase space into their level sets.

Accepted Answer

Probably the most widespread method for visualizing phase space structure is simply plotting the trajectories associated with as many initial conditions, and for as long a time, as possible.

Accepted Answer

Their algorithm consists of making a density or contour plot of the time averages of functions in a two-dimensional slice of the phase space.

Accepted Answer

The authors consider partitions of the phase space A , on which a dynamical system T that preserves a smooth measure m on A acts, into invariant sets induced by functions that are constant on the orbits of the dynamical system.

Accepted Answer

The ergodic partition of a compact metric space A , under the dynamics of a continuous automorphism T , is shown to be the product of measurable partitions of the space induced by the time averages of a set of functions on A .

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Since the different invariant sets that are identified are ergodic, a statistical theory of mixing on each component would then follow immediately.

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In his study, he looks at the behavior of one orbit in a positive area ergodic component of the Standard Map and finds that only after times as long as 1010 iterations does the orbit closely exhibit ergodic behavior.

Accepted Answer

Observe that the set of all restrictions of functions in C(A) to C , denoted C(A)uC forms a dense set in the set of all integrable ~with respect to mC! functions on C .

Accepted Answer

The authors have shown that ergodic partition can be constructed out of sets on which the time-averages of a dense, countable set of functions in the space of all continuous functions are constant.

Accepted Answer

The ABC map is a map on a threedimensional torus that was studied to provide an example of the application of their algorithm to maps with a smooth invariant measure in dimensions bigger than two, where plotting the orbits of a map on a screen often results in a scatter of points that is not convenient for analysis.

Accepted Answer

Redistribution subject to Apreserving ABC map introduced by Feingold et al. and shown that their method of visualization produces good contrast in reasonable iteration time.

Accepted Answer

Vol. 9, No. 1, 1999 I. Mezić and S. WigginsDosin* f in~x !5 1 n (i50 N~y ,z !21 sin~xi!,and N(y ,z) is the number of iterates it takes for the finite time average of sin x to converge starting from the point (y ,z) in the x50.01 plane.

A method for visualization of invariant sets of dynamical systems based on the ergodic partition

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Most frequently asked questions

1. What contributions have the authors mentioned in the paper "A method for visualization of invariant sets of dynamical systems based on the ergodic partition" ?

2. What have the authors stated for future works in "A method for visualization of invariant sets of dynamical systems based on the ergodic partition" ?

3. What is the common method for the visualization of phase space structure in four-dimensional dynamical?

4. What is the meaning of ergodic partitions?

5. What are the issues that the authors only touched upon here?

6. What is the method for plotting exit times?

7. What is the numerical algorithm for partitioning phase space?

8. What is the common method for visualizing phase space structure?

9. What is the purpose of the algorithm?

10. what is the ergodic partition of the phase space?

11. What is the ergodic partition of a compact metric space?

12. what is the ergodic theory of mixing?

13. How many iterations does the orbit exhibit ergodic behavior?

14. What is the proof that mC is an ergodic measure?

15. What is the way to show that ergodic partition can be constructed out of sets?

16. What is the definition of the ABC map?

17. What is the way to visualize the ABC map?

18. what is the g in a x50.01 plane?

Figures

Citations

Koopman Operator Spectrum for Random Dynamical Systems

A method for revealing phase space structures of general time dependent dynamical systems

Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stability

A Theoretical Framework for Lagrangian Descriptors

Transport and stirring induced by vortex formation

References

Ergodic theory and differentiable dynamics

Zur Operatorenmethode In Der Klassischen Mechanik

On the geometrical and statistical properties of dynamical systems : theory and applications

The decomposition of measures

On A Theorem of Dieudonné.

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