Enrique R. Pujals enrique@impa · different backgrounds... Robust and generic dynamics Enrique R....

Transcript of Enrique R. Pujals enrique@impa · different backgrounds... Robust and generic dynamics Enrique R....

different backgrounds...

Robust and generic dynamics

Enrique R. [email protected]

May 2011, SIAM

Enrique R. Pujals (IMPA) Dynamical Systems SIAM 2011 1 / 29

Setting

Continuous dynamics

ẋ = F (x)

t → Φt (x)

Discrete dynamics

Smooth map, f : M → M, where M is a smooth compact manifold

x f (x) f 2(x) · · · f n(x) f n+1(x)


Setting

Continuous dynamics

ẋ = F (x)

t → Φt (x)

Discrete dynamics


x f (x) f 2(x) · · · f n(x) f n+1(x)


Setting

Continuous dynamics

ẋ = F (x)

t → Φt (x)

Discrete dynamics


x f (x) f 2(x) · · · f n(x) f n+1(x)


Setting

Continuous dynamics

ẋ = F (x)

t → Φt (x)

Discrete dynamics


x f (x) f 2(x) · · · f n(x) f n+1(x)


Setting

Continuous dynamics

ẋ = F (x)

t → Φt (x)

Discrete dynamics


x f (x) f 2(x) · · · f n(x) f n+1(x)


Setting

Continuous dynamics

ẋ = F (x)

t → Φt (x)

Discrete dynamics


x f (x) f 2(x) · · · f n(x) f n+1(x)


Setting

Continuous dynamics

ẋ = F (x)

t → Φt (x)

Discrete dynamics


x f (x) f 2(x) · · · f n(x) f n+1(x)


Goal

We can “compute the trajectories“ “to describe the dynamics”

Existence of periodic trajectories?Assymptotic behavior of the trajectories? attractors?“chaotic”?

Can we trust what we get from numerical approximation?

How do we interpret the numerical solutions?


Goal






Goal


Existence of periodic trajectories?

Assymptotic behavior of the trajectories? attractors?“chaotic”?




Goal


Existence of periodic trajectories?Assymptotic behavior of the trajectories? attractors?

“chaotic”?




Goal






Goal






Goal






A taxonomy could help

“A CLASSIFICATION of possible dynamics”would help to interpret the results

A “TAXONOMY“ of generic well described dynamical behaviors


“How to get a Taxonomy”

Looking for the possible generic dynamical scenario

“MODEL OF THE MODELS”

“Reality to be understood“ Mathematical Models.

Dynamical Equations Model of those Mathematical Models

(C. Jones) in theory in delirium (?)

Properties requiere to those Model of models:

DYNAMICALLY WELL DESCRIBED.GENERIC.Easy to treat analytically, built “geometrically”.





“Reality to be understood“

Mathematical Models.










Dynamical Equations

Model of those Mathematical Models










(C. Jones)

in theory in delirium (?)









(C. Jones) in theory

in delirium (?)











DYNAMICALLY WELL DESCRIBED.

GENERIC.Easy to treat analytically, built “geometrically”.









DYNAMICALLY WELL DESCRIBED.GENERIC.

Easy to treat analytically, built “geometrically”.









DYNAMICALLY WELL DESCRIBED.GENERIC.Easy to treat analytically,

built “geometrically”.


How to get those model of the models? Example.

Poincaré and 3 body problem3 bodies interacting under their gravitational attraction.

2 body problem/Keplerian problem.

Elliptic motions. Motions are periodic.

Poincaré: 3 body problem as a perturbation of the 2 body one.



Poincaré and 3 body problem

3 bodies interacting under their gravitational attraction.








Elliptic motions.

Motions are periodic.



3 body problem.

Poincaré observed “complicated trajectories”.

“...One will be struck by the complexity of this figure, which I am noteven attempting to draw.”

“...Nothing can give us a better idea of the intricacy of the three-bodyproblem....”


3 body problem.





3 body problem.





3 body problem.





3 body problem.





How is that possible?

Find a DYNAMICAL CONFIGURATION:

that provides the dynamical properties that we want;gives more information;

it is easy to be detected;it is “UNIVERSAL”.

It can be found in the original problem.




that provides the dynamical properties that we want;

gives more information;







it is easy to be detected;

it is “UNIVERSAL”.



Poincaré: Homoclinic points.

Homoclinic points: “Points with same past and future"




Infinitely many periodic orbits.





Chaotic dynamic.





Chaotic dynamic.

It holds in a robust way.


Homoclic points.

Appears in the 3 body problem.

Perturbed pendulum.

Generic in mechanical problems with at least two degree of freedom

DichotomyEither the dynamics is VERY SIMPLE: Integrable. Morse-Smale inthe non-conservative case,

or there is a transversal homoclinic point. There are chaoticcomponents.


Homoclic points.


Perturbed pendulum.





Homoclic points.


Perturbed pendulum.





Homoclic points.


Perturbed pendulum.





Homoclic points.


Perturbed pendulum.


Dichotomy

Either the dynamics is VERY SIMPLE: Integrable. Morse-Smale inthe non-conservative case,



Homoclic points.


Perturbed pendulum.


DichotomyEither the dynamics is VERY SIMPLE:

Integrable. Morse-Smale inthe non-conservative case,



Homoclic points.


Perturbed pendulum.


DichotomyEither the dynamics is VERY SIMPLE: Integrable.

Morse-Smale inthe non-conservative case,



Homoclic points.


Perturbed pendulum.





Homoclic points.


Perturbed pendulum.





Homoclic points.


Perturbed pendulum.





Homoclic points.


Perturbed pendulum.



or there is a transversal homoclinic point.

There are chaoticcomponents.


Homoclic points.


Perturbed pendulum.





Transv. Hom. points: source of Hyperbolic theory.

Smale’s Horseshoes:

Hyperbolic Structure

TΛM = Es ⊕ Eu

||Df/Es || < λ < 1, (squeeze) ||Df−1/Eu || < λ < 1, (strecht)





TΛM = Es ⊕ Eu






TΛM = Es ⊕ Eu






TΛM = Es ⊕ Eu






TΛM = Es ⊕ Eu






TΛM = Es ⊕ Eu






TΛM = Es ⊕ Eu



Nice description of Hyperbolic systems

ROBUST CHAOTIC dynamics.

ROBUST TRANSITIVE dynamics.

1 Infinitely many periodic orbits,2 Stable systems;3 Nice statistical properties.

NICE PICTURE FOR HYPERBOLIC SYSTEMS

Hyperbolicity, a happy story.





1 Infinitely many periodic orbits,

2 Stable systems;3 Nice statistical properties.







1 Infinitely many periodic orbits,2 Stable systems;

3 Nice statistical properties.




Mechanical examples of Hyperbolic systems

The triple linkage

Geodesic flow in a manifold of negative curvature (Hunt-MacKay).

GLOBAL HYPERBOLIC FLOW.


Mechanical examples of Hyperbolic systemsThe triple linkage




”Robust” implies Hyperbolicity?

There are robust dynamics which are not hyperbolic

Only in higher dimension

For surfaces: Robust chaotic dynamics —-> Hyperbolicity.





For surfaces: Robust chaotic dynamics

—-> Hyperbolicity.


Non-Hyperbolic Robust transitive systems.

1 take f : M2 → M2 Hyperbolic diffeo;2 take g : N2 → N2 be either an integrable one (pendulum/forced

pendulum);

f × g : M2 × N2 → M2 × N2.

PARTIAL HYPERBOLICITY

Arnold’s diffusion



1 take f : M2 → M2 Hyperbolic diffeo;

2 take g : N2 → N2 be either an integrable one (pendulum/forcedpendulum);

f × g : M2 × N2 → M2 × N2.






pendulum);

f × g : M2 × N2 → M2 × N2.






pendulum);

f × g : M2 × N2 → M2 × N2.






pendulum);

f × g : M2 × N2 → M2 × N2.






pendulum);

f × g : M2 × N2 → M2 × N2.






pendulum);

f × g : M2 × N2 → M2 × N2.






pendulum);

f × g : M2 × N2 → M2 × N2.




Robust dynamic for flows: Lorenz Attractor

X (x , y , z) =

ẋ = −αx + αyẏ = βx − y − xzż = −γz + xy ,

Lorenz: for parameters close to 10,28,8/3 there is a RobustTransitive Attractor containing a singularity (equilibrium point).

Presences of singularity prevents the hyperbolicity



X (x , y , z) =






X (x , y , z) =






X (x , y , z) =





Geometric Lorenz attractors

Geometric models of robust transitive attractor with singularities(Guckenheimer-Williams).


Singular Hyperbolicity

Structure underlying the GLA

TΛ = Es ⊕ Ecu,

Rich dynamic obtained from Singular hyperbolicity:1 Density of periodic orbits/ chaoticity;2 Good statistical properties.

Lorenz’s equation is a Geometric Lorenz Attractor

Any C1−robust attractor with singularitiesis a Geometric Lorenz Attractor.




TΛ = Es ⊕ Ecu,







TΛ = Es ⊕ Ecu,







TΛ = Es ⊕ Ecu,

Rich dynamic obtained from Singular hyperbolicity:

1 Density of periodic orbits/ chaoticity;2 Good statistical properties.






TΛ = Es ⊕ Ecu,

Rich dynamic obtained from Singular hyperbolicity:1 Density of periodic orbits/ chaoticity;

2 Good statistical properties.






TΛ = Es ⊕ Ecu,







TΛ = Es ⊕ Ecu,







TΛ = Es ⊕ Ecu,





Robustness implies structure.

Can we characterize robust dynamics which are not hyperbolic

ROBUSTNESS IMPLIES WEAK HYPERBOLICITY.

NICE DYNAMICAL PROPERTIES (sometimes under extra assumptions)

Abundance of periodic trajectories;good statistical properties.




ROBUSTNESS IMPLIES

WEAK HYPERBOLICITY.








Abundance of periodic trajectories;

good statistical properties.


Everything is ”Robust”?

NO. WILD DYNAMICS.

Infinitely many attractors apearing and dissapearing



NO. WILD DYNAMICS.




NO. WILD DYNAMICS.




NO. WILD DYNAMICS.




NO. WILD DYNAMICS.




NO. WILD DYNAMICS.


Perturbations of Integrable systems



NO. WILD DYNAMICS.


Perturbations of Integrable systems

New kind of homoclinic pointsEnrique R. Pujals (IMPA) Dynamical Systems SIAM 2011 20 / 29

Homoclinic points: Wild dynamics

Homoclinic points:

Hyperbolicity=Transversal homoclinic points.

Breaking Hyperbolicity =Tangent homoclinic points.



Homoclinic points:





Homoclinic points:




TANGENCIES IMPLY WILD SETS

Newhouse: Unfolding homoclinic tangencies

Residual sets of diffeos having infinitely many periodic attractors.

One tangency —> a cascade of infinitely many tangencies —> infinitelymany periodic attractor.





One tangency —>

a cascade of infinitely many tangencies —> infinitelymany periodic attractor.





One tangency —> a cascade of infinitely many tangencies —>

infinitelymany periodic attractor.

different backgrounds...Enrique R. Pujals (IMPA) Dynamical Systems SIAM 2011 23 / 29

GENERIC DYNAMICS/UNIVERSAL MECHANISMS

Coming back to the possible dynamical scenario:

Robust dynamics also called Tame dynamics

Wild dynamics

How can we explain those dynamical behaviors?

Dictionary?

Phenomenas/Mechanisms

Phenomena: Some well described dynamical behavior

Mechanisms: Some dynamical configuration and its perturbations.





Wild dynamics


Dictionary?







Robust dynamics

also called Tame dynamics

Wild dynamics


Dictionary?








Wild dynamics


Dictionary?








Wild dynamics


Dictionary?








Wild dynamics


Dictionary?








Wild dynamics


Dictionary?








Wild dynamics


Dictionary?








Wild dynamics


Dictionary?








Wild dynamics


Dictionary?





UNIVERSAL MECHANISMS/GENERIC DYNAMICS

PHENOMENA MECHANISMS

The collection of phenomenas has to be generic

The collection of mechanisms has to be simply and generic

Rich in their consequences

“HOMOCLINIC BIFURCATIONS should be the MECHANISMS”


Dichotomy in low dimension.

Dynamical dichotomy for low dimension:

Robust Dynamics = Hyperbolic systems

Wild dynamic “=” homoclinic tangencies.


What about higher dimensions?

The situation changes dramatically!!

The ZOO (phenomenas) is becomes larger.

There are Robust chaotic dynamics which are NOT hyperbolic.

New type of wild dynamics.

New Mechanisms/Homoclinic bifurcations get into the picture.

Heterodimensional cycle.


Phenom.Mechanisms/ Dichotomies in any dimen?

There is reasonable (vague), almost generic list of phenomenas.

There is reasonable list of mechanisms.

Some relation between New phenomenas/New mechanisms

Incomplete dictionary


Phenom.Mechanisms/ Dictionari in any dimen?


Phenom.Mechanisms/ Dictionari in any dimen?

Enrique R. Pujals enrique@impa · different backgrounds... Robust and generic dynamics Enrique R....

Documents

Transcript of Enrique R. Pujals enrique@impa · different backgrounds... Robust and generic dynamics Enrique R....