Geometric Dynamic Days Heidelberg Germany Oct 27 28
Geometric Dynamic Days Heidelberg, Germany Oct 27 -28, 2017 Lecture 2 : Shear-induced Chaos Lai-Sang Young Courant Institute, NYU http: //www. cims. nyu. edu/~lsy/
A theory of strange attractors (1) SRB measures and observable chaos (2) Shear-induced chaos -- main class of examples outside of the Axiom A category Setting = Riemannian manifold or region of embedding with Call the attractor and Not a formal definition, but think of , cpct closure the basin of attraction as “chaotic” ( e. g. more complicated than fixed pt or period orbit)
A theory of strange attractors (1) SRB measures and observable chaos (2) Shear-induced chaos -- main class of examples outside of the Axiom A category Setting = Riemannian manifold or region of embedding with Call the attractor and Not a formal definition, but think of , cpct closure the basin of attraction as “chaotic” ( e. g. more complicated than fixed pt or period orbit)
A theory of strange attractors (1) SRB measures and observable chaos (2) Shear-induced chaos -- main class of examples outside of the Axiom A category Setting = Riemannian manifold or region of embedding with Call the attractor and Not a formal definition, but think of , cpct closure the basin of attraction as “chaotic” ( e. g. more complicated than fixed pt or period orbit)
An often adopted point of view is observable events = positive Leb measure sets For Hamiltonian systems, Liouville measure = the important invariant measure Same for volume preserving dynamical systems But what about “dissipative” systems, e. g. , one with an attractor ? Consider Assume f is volume decreasing. Then , and all inv meas are supported on i. e. , no inv meas has a density wrt Leb Proof: . . .
An often adopted point of view is observable events = positive Leb measure sets For Hamiltonian systems, Liouville measure = the important invariant measure Same for volume preserving dynamical systems But what about “dissipative” systems, e. g. , one with an attractor ? Consider Assume f is volume decreasing. Then , and all inv meas are supported on i. e. , no inv meas has a density wrt Leb Proof: . . .
An often adopted point of view is observable events = positive Leb measure sets For Hamiltonian systems, Liouville measure = the important invariant measure Same for volume preserving dynamical systems But what about “dissipative” systems, e. g. , one with an attractor ? Consider Assume f is volume decreasing. Then , and all inv meas are supported on i. e. , no inv meas has a density wrt Leb Proof: . . .
An often adopted point of view is observable events = positive Leb measure sets For Hamiltonian systems, Liouville measure = the important invariant measure Same for volume preserving dynamical systems But what about “dissipative” systems, e. g. , one with an attractor ? Consider Assume f is volume decreasing. Then , and all inv meas are supported on i. e. , no inv meas has a density wrt Leb Proof: . . .
An often adopted point of view is observable events = positive Leb measure sets For Hamiltonian systems, Liouville measure = the important invariant measure Same for volume preserving dynamical systems But what about “dissipative” systems, e. g. , one with an attractor ? Consider Assume f is volume decreasing. Then , and all inv meas are supported on i. e. , no inv meas has a density wrt Leb Proof: . . .
If for attractors all invariant measures are singular wrt Leb, - how can any one of them be physically relevant, - and what use is ergodic theory for studying observable events? Not so fast. . (1) Fixed point sink : Leb a. e. point converges to sink (2) Uniformly hyperbolic (or Axiom A) attractors : SRB measures Sinai, Ruelle, Bowen in 1970 s constructed a special inv meas supported on attractor but with the property that for every continuous ______ Note: Not the Birkhoff Ergodic Theorem BET says if is ergodic, then time avg = space avg for ---
If for attractors all invariant measures are singular wrt Leb, - how can any one of them be physically relevant, - and what use is ergodic theory for studying observable events? Not so fast. . (1) Fixed point sink : Leb a. e. point converges to sink (2) Uniformly hyperbolic (or Axiom A) attractors : SRB measures Sinai, Ruelle, Bowen in 1970 s constructed a special inv meas supported on attractor but with the property that for every continuous ______ Note: Not the Birkhoff Ergodic Theorem BET says if is ergodic, then time avg = space avg for ---
If for attractors all invariant measures are singular wrt Leb, - how can any one of them be physically relevant, - and what use is ergodic theory for studying observable events? Not so fast. . (1) Fixed point sink : Leb a. e. point converges to sink (2) Uniformly hyperbolic (or Axiom A) attractors : SRB measures Sinai, Ruelle, Bowen in 1970 s constructed a special inv meas supported on attractor but with the property that for every continuous ______ Note: Not the Birkhoff Ergodic Theorem BET says if is ergodic, then time avg = space avg for ---
If for attractors all invariant measures are singular wrt Leb, - how can any one of them be physically relevant, - and what use is ergodic theory for studying observable events? Not so fast. . (1) Fixed point sink : Leb a. e. point converges to sink (2) Uniformly hyperbolic (or Axiom A) attractors : SRB measures Sinai, Ruelle, Bowen in 1970 s constructed a special inv meas supported on attractor but with the property that for every continuous ______ Note: Not the Birkhoff Ergodic Theorem BET says if is ergodic, then time avg = space avg for ---
A uniformly hyperbolic or Axiom A attractor is one on which Df has saddle-like behavior at every point , i. e. Fact: Such attractors are unions of unstable manifolds, i. e. and their basins are foliated by stable manifolds , i. e. How SRB measures work: -- these invariant measures have densities on W^u -- If then -- integrating out along W^s, measure properties passed along to points in the basin Uses absolute continuity of -foliation.
A uniformly hyperbolic or Axiom A attractor is one on which Df has saddle-like behavior at every point , i. e. Fact: Such attractors are unions of unstable manifolds, i. e. and their basins are foliated by stable manifolds , i. e. How SRB measures work: -- these invariant measures have densities on W^u -- If then -- integrating out along W^s, measure properties passed along to points in the basin Uses absolute continuity of -foliation.
A uniformly hyperbolic or Axiom A attractor is one on which Df has saddle-like behavior at every point , i. e. Fact: Such attractors are unions of unstable manifolds, i. e. and their basins are foliated by stable manifolds , i. e. How SRB measures work: -- these invariant measures have densities on W^u -- If then -- integrating out along W^s, measure properties passed along to points in the basin Uses absolute continuity of -foliation.
A uniformly hyperbolic or Axiom A attractor is one on which Df has saddle-like behavior at every point , i. e. Fact: Such attractors are unions of unstable manifolds, i. e. and their basins are foliated by stable manifolds , i. e. How SRB measures work: -- these invariant measures have densities on W^u -- If then -- integrating out along W^s, measure properties passed along to points in the basin Uses absolute continuity of -foliation.
A uniformly hyperbolic or Axiom A attractor is one on which Df has saddle-like behavior at every point , i. e. Fact: Such attractors are unions of unstable manifolds, i. e. and their basins are foliated by stable manifolds , i. e. How SRB measures work: -- these invariant measures have densities on W^u -- If then -- integrating out along W^s, measure properties passed along to points in the basin Uses absolute continuity of -foliation.
NO Axiom A General smooth ergodic theory assumption is a diffeomorphism where Consider and is a Borel probability invariant measure Lyapunov exponents are defined Say is nonuniformly hyperbolic if it has positive & negative Lyap exp Here defined but no lower bound on angles and onset of exp/contr not uniform and defined but no lower bound in length or upper bound in curvature Theorem [Pugh-Shub 1989]Absolute continuity of -foliation Note given f , usually (uncountably) many invariant measures. Definition is called an SRB measure if -- f has a positive Lyapunov exp - a. e. -- conditional measures of on -manifolds are smooth
NO Axiom A General smooth ergodic theory assumption is a diffeomorphism where Consider and is a Borel probability invariant measure Lyapunov exponents are defined Say is nonuniformly hyperbolic if it has positive & negative Lyap exp Here defined but no lower bound on angles and onset of exp/contr not uniform and defined but no lower bound in length or upper bound in curvature Theorem [Pugh-Shub 1989]Absolute continuity of -foliation Note given f , usually (uncountably) many invariant measures. Definition is called an SRB measure if -- f has a positive Lyapunov exp - a. e. -- conditional measures of on -manifolds are smooth
NO Axiom A General smooth ergodic theory assumption is a diffeomorphism where Consider and is a Borel probability invariant measure Lyapunov exponents are defined Say is nonuniformly hyperbolic if it has positive & negative Lyap exp Here defined but no lower bound on angles and onset of exp/contr not uniform and defined but no lower bound in length or upper bound in curvature Theorem [Pugh-Shub 1989]Absolute continuity of -foliation Note given f , usually (uncountably) many invariant measures. Definition is called an SRB measure if -- f has a positive Lyapunov exp - a. e. -- conditional measures of on -manifolds are smooth
NO Axiom A General smooth ergodic theory assumption is a diffeomorphism where Consider and is a Borel probability invariant measure Lyapunov exponents are defined Say is nonuniformly hyperbolic if it has positive & negative Lyap exp Here defined but no lower bound on angles and onset of exp/contr not uniform and defined but no lower bound in length or upper bound in curvature Theorem [Pugh-Shub 1989]Absolute continuity of -foliation Note given f , usually (uncountably) many invariant measures. Definition is called an SRB measure if -- f has a positive Lyapunov exp - a. e. -- conditional measures of on -manifolds are smooth
NO Axiom A General smooth ergodic theory assumption is a diffeomorphism where Consider and is a Borel probability invariant measure Lyapunov exponents are defined Say is nonuniformly hyperbolic if it has positive & negative Lyap exp Here defined but no lower bound on angles and onset of exp/contr not uniform and defined but no lower bound in length or upper bound in curvature Theorem [Pugh-Shub 1989]Absolute continuity of -foliation Note given f , usually (uncountably) many invariant measures. Definition is called an SRB measure if -- f has a positive Lyapunov exp - a. e. -- conditional measures of on -manifolds are smooth
Theorem If then is an ergodic, nonunif hyp SRB meas, for all set. in a pos Leb meas (same proof, less tidy) Fact : Some other properties of SRB meas in unif hyp setting extend to nonunif setting e. g. entropy formula[Ledrappier-Strelcyn, L, L-Young 1980 s] but no claim of existence for “chaotic-looking attractors” A generalization of uniform hyperbolicity is existence of invariant cones ( = separation of relative expansion & contraction) In the absence of invariant cones (or uniform hyperbolicity), proving existence of SRB measure or the positivity of Lyap exp is a major challenge even for systems with a great deal of instability Reason : where there is expansion, there is also contraction. . = tangent vector at x, sometimes grows, sometimes shrinkscancellation can be delicate
Theorem If then is an ergodic, nonunif hyp SRB meas, for all set. in a pos Leb meas (same proof, less tidy) Fact : Some other properties of SRB meas in unif hyp setting extend to nonunif setting e. g. entropy formula[Ledrappier-Strelcyn, L, L-Young 1980 s] but no claim of existence for “chaotic-looking attractors” A generalization of uniform hyperbolicity is existence of invariant cones ( = separation of relative expansion & contraction) In the absence of invariant cones (or uniform hyperbolicity), proving existence of SRB measure or the positivity of Lyap exp is a major challenge even for systems with a great deal of instability Reason : where there is expansion, there is also contraction. . = tangent vector at x, sometimes grows, sometimes shrinkscancellation can be delicate
Theorem If then is an ergodic, nonunif hyp SRB meas, for all set. in a pos Leb meas (same proof, less tidy) Fact : Some other properties of SRB meas in unif hyp setting extend to nonunif setting e. g. entropy formula[Ledrappier-Strelcyn, L, L-Young 1980 s] but no claim of existence for “chaotic-looking attractors” A generalization of uniform hyperbolicity is existence of invariant cones ( = separation of relative expansion & contraction) In the absence of invariant cones (or uniform hyperbolicity), proving existence of SRB measure or the positivity of Lyap exp is a major challenge even for systems with a great deal of instability Reason : where there is expansion, there is also contraction. . = tangent vector at x, sometimes grows, sometimes shrinkscancellation can be delicate
Theorem If then is an ergodic, nonunif hyp SRB meas, for all set. in a pos Leb meas (same proof, less tidy) Fact : Some other properties of SRB meas in unif hyp setting extend to nonunif setting e. g. entropy formula[Ledrappier-Strelcyn, L, L-Young 1980 s] but no claim of existence for “chaotic-looking attractors” A generalization of uniform hyperbolicity is existence of invariant cones ( = separation of relative expansion & contraction) In the absence of invariant cones (or uniform hyperbolicity), proving existence of SRB measure or the positivity of Lyap exp is a major challenge even for systems with a great deal of instability Reason : where there is expansion, there is also contraction. . = tangent vector at x, sometimes grows, sometimes shrinkscancellation can be delicate
Theorem If then is an ergodic, nonunif hyp SRB meas, for all set. in a pos Leb meas (same proof, less tidy) Fact : Some other properties of SRB meas in unif hyp setting extend to nonunif setting e. g. entropy formula[Ledrappier-Strelcyn, L, L-Young 1980 s] but no claim of existence for “chaotic-looking attractors” A generalization of uniform hyperbolicity is existence of invariant cones ( = separation of relative expansion & contraction) In the absence of invariant cones (or uniform hyperbolicity), proving existence of SRB measure or the positivity of Lyap exp is a major challenge even for systems with a great deal of instability Reason : where there is expansion, there is also contraction. . = tangent vector at x, sometimes grows, sometimes shrinkscancellation can be delicate
Theorem If then is an ergodic, nonunif hyp SRB meas, for all set. in a pos Leb meas (same proof, less tidy) Fact : Some other properties of SRB meas in unif hyp setting extend to nonunif setting e. g. entropy formula[Ledrappier-Strelcyn, L, L-Young 1980 s] but no claim of existence for “chaotic-looking attractors” A generalization of uniform hyperbolicity is existence of invariant cones ( = separation of relative expansion & contraction) In the absence of invariant cones (or uniform hyperbolicity), proving existence of SRB measure or the positivity of Lyap exp is a major challenge even for systems with a great deal of instability Reason : where there is expansion, there is also contraction. . = tangent vector at x, sometimes grows, sometimes shrinkscancellation can be delicate
cancellation can be delicate. . and the problem is real Recall: Example (linear shear flow) kicks delivered at period T assumin g key : Theorem (Wang-Young, 02) (a) small : invariant closed curve (b) as develop (c) large increases : invariant curve breaks, horseshoes and : mixed behavior sustained, observable chaos on positive meas set horseshoes + sinks on open set
cancellation can be delicate. . and the problem is real Recall: Example (linear shear flow) kicks delivered at period T assumin g key : Theorem (Wang-Young, 02) (a) small : invariant closed curve (b) as develop (c) large increases : invariant curve breaks, horseshoes and : mixed behavior sustained, observable chaos on positive meas set horseshoes + sinks on open set
cancellation can be delicate. . and the problem is real Recall: Example (linear shear flow) kicks delivered at period T assumin g key : Theorem (Wang-Young, 02) (a) small : invariant closed curve (b) as develop (c) large increases : invariant curve breaks, horseshoes and : mixed behavior sustained, observable chaos on positive meas set horseshoes + sinks on open set
cancellation can be delicate. . and the problem is real Recall: Example (linear shear flow) kicks delivered at period T assumin g key : Theorem (Wang-Young, 02) (a) small : invariant closed curve (b) as develop (c) large increases : invariant curve breaks, horseshoes and : mixed behavior sustained, observable chaos on positive meas set horseshoes + sinks on open set
Main class of attractors with no inv cones known to possess SRB meas are phenomenologically : shear induced chaos technical math : “rank one” ( = one unstable direction) obtained by applying the following generic result Theorem Setting : [Wang-Young 2002, 2008] where = parameter, [technical details omitted] (m-dim disk) m = “determinant” (dissipation) Assume 1. singular limit defined, i. e. 2. has “enough expansion” 3. nondegeneracy + transversality conditions Then for all suff small , = pos meas set of s. t. (a) has an ergodic SRB measure (b) Leb-a. e. in M Precursor to theorem above is: the Henon maps [Benedicks-Carleson 1990, B-Young 1993]
Main class of attractors with no inv cones known to possess SRB meas are phenomenologically : shear induced chaos technical math : “rank one” ( = one unstable direction) obtained by applying the following generic result Theorem Setting : [Wang-Young 2002, 2008] where = parameter, [technical details omitted] (m-dim disk) m = “determinant” (dissipation) Assume 1. singular limit defined, i. e. 2. has “enough expansion” 3. nondegeneracy + transversality conditions Then for all suff small , = pos meas set of s. t. (a) has an ergodic SRB measure (b) Leb-a. e. in M Precursor to theorem above is: the Henon maps [Benedicks-Carleson 1990, B-Young 1993]
Main class of attractors with no inv cones known to possess SRB meas are phenomenologically : shear induced chaos technical math : “rank one” ( = one unstable direction) obtained by applying the following generic result Theorem Setting : [Wang-Young 2002, 2008] where = parameter, [technical details omitted] (m-dim disk) m = “determinant” (dissipation) Assume 1. singular limit defined, i. e. 2. has “enough expansion” 3. nondegeneracy + transversality conditions Then for all suff small , = pos meas set of s. t. (a) has an ergodic SRB measure (b) Leb-a. e. in M Precursor to theorem above is: the Henon maps [Benedicks-Carleson 1990, B-Young 1993]
Main class of attractors with no inv cones known to possess SRB meas are phenomenologically : shear induced chaos technical math : “rank one” ( = one unstable direction) obtained by applying the following generic result Theorem Setting : [Wang-Young 2002, 2008] where = parameter, [technical details omitted] (m-dim disk) m = “determinant” (dissipation) Assume 1. singular limit defined, i. e. 2. has “enough expansion” 3. nondegeneracy + transversality conditions Then for all suff small , = pos meas set of s. t. (a) has an ergodic SRB measure (b) Leb-a. e. in M Precursor to theorem above is: the Henon maps [Benedicks-Carleson 1990, B-Young 1993]
Main class of attractors with no inv cones known to possess SRB meas are phenomenologically : shear induced chaos technical math : “rank one” ( = one unstable direction) obtained by applying the following generic result Theorem Setting : [Wang-Young 2002, 2008] where = parameter, [technical details omitted] (m-dim disk) m = “determinant” (dissipation) Assume 1. singular limit defined, i. e. 2. has “enough expansion” 3. nondegeneracy + transversality conditions Then for all suff small , = pos meas set of s. t. (a) has an ergodic SRB measure (b) Leb-a. e. in M Precursor to theorem above is: the Henon maps [Benedicks-Carleson 1990, B-Young 1993]
Main class of attractors with no inv cones known to possess SRB meas are phenomenologically : shear induced chaos technical math : “rank one” ( = one unstable direction) obtained by applying the following generic result Theorem Setting : [Wang-Young 2002, 2008] where = parameter, [technical details omitted] (m-dim disk) m = “determinant” (dissipation) Assume 1. singular limit defined, i. e. 2. has “enough expansion” 3. nondegeneracy + transversality conditions Then for all suff small , = pos meas set of s. t. (a) has an ergodic SRB measure (b) Leb-a. e. in M Precursor to theorem above is: the Henon maps [Benedicks-Carleson 1990, B-Young 1993]
Rest of lecture: generalization of 2 D linear shear flow example Kicking of general limit cycles = smooth flow with hyperbolic (attracting) limit cycle , period basin of attraction kick map , kick period , Assume , so attractor well defined. strong stable mfld for Here so For where multiple of , and is the sliding map along To connect with rank-one theorem, -leaves. singular limit as .
Rest of lecture: generalization of 2 D linear shear flow example Kicking of general limit cycles = smooth flow with hyperbolic (attracting) limit cycle , period basin of attraction kick map , kick period , Assume , so attractor well defined. strong stable mfld for Here so For where multiple of , and is the sliding map along To connect with rank-one theorem, -leaves. singular limit as .
Rest of lecture: generalization of 2 D linear shear flow example Kicking of general limit cycles = smooth flow with hyperbolic (attracting) limit cycle , period basin of attraction kick map , kick period , Assume , so attractor well defined. strong stable mfld for Here so For where multiple of , and is the sliding map along To connect with rank-one theorem, -leaves. singular limit as .
Rest of lecture: generalization of 2 D linear shear flow example Kicking of general limit cycles = smooth flow with hyperbolic (attracting) limit cycle , period basin of attraction kick map , kick period , Assume , so attractor well defined. strong stable mfld for Here so For where multiple of , and is the sliding map along To connect with rank-one theorem, -leaves. singular limit as .
Rest of lecture: generalization of 2 D linear shear flow example Kicking of general limit cycles = smooth flow with hyperbolic (attracting) limit cycle , period basin of attraction kick map , kick period , Assume , so attractor well defined. strong stable mfld for Here so For where multiple of , and is the sliding map along To connect with rank-one theorem, -leaves. singular limit as .
Note not all kicks are equally effective in producing chaotic behavior e. g. kicks roughly || -leaves or kicks that carries one -leaf lead to no folding of . For general limit cycles, cannot separate quantities in Think of shear-induced chaos as stretching + folding caused by variation in the action of on relative to Linear shear flow in n-dimensions where nonzero matrix w/ Re(ev)>0 For this system, -leaves are hyperplanes orthogonal to What determines dynamics: variation of amplitude of kick direction of kick
Note not all kicks are equally effective in producing chaotic behavior e. g. kicks roughly || -leaves or kicks that carries one -leaf lead to no folding of . For general limit cycles, cannot separate quantities in Think of shear-induced chaos as stretching + folding caused by variation in the action of on relative to Linear shear flow in n-dimensions where nonzero matrix w/ Re(ev)>0 For this system, -leaves are hyperplanes orthogonal to What determines dynamics: variation of amplitude of kick direction of kick
Note not all kicks are equally effective in producing chaotic behavior e. g. kicks roughly || -leaves or kicks that carries one -leaf lead to no folding of . For general limit cycles, cannot separate quantities in Think of shear-induced chaos as stretching + folding caused by variation in the action of on relative to Linear shear flow in n-dimensions where nonzero matrix w/ Re(ev)>0 For this system, -leaves are hyperplanes orthogonal to What determines dynamics: variation of amplitude of kick direction of kick
Note not all kicks are equally effective in producing chaotic behavior e. g. kicks roughly || -leaves or kicks that carries one -leaf lead to no folding of . For general limit cycles, cannot separate quantities in Think of shear-induced chaos as stretching + folding caused by variation in the action of on relative to Linear shear flow in n-dimensions where nonzero matrix w/ Re(ev)>0 For this system, -leaves are hyperplanes orthogonal to What determines dynamics: variation of amplitude of kick direction of kick
Another version of the same idea: Generic supercritical Hopf bifurcations i. e. pair of cx eigenvalues cross Imaxis at = limit cycle, radius Normal form : generic : Introduce twist number Theorem. (2 D) If we apply with period T a kick such that is large enough , then for large enough T, a positive parameter set of flow-maps have an ergodic SRB measure that controls the large-time dynamics of Leb-a. e. point in the basin. [Wang-Young 2003]
Another version of the same idea: Generic supercritical Hopf bifurcations i. e. pair of cx eigenvalues cross Imaxis at = limit cycle, radius Normal form : generic : Introduce twist number Theorem. (2 D) If we apply with period T a kick such that is large enough , then for large enough T, a positive parameter set of flow-maps have an ergodic SRB measure that controls the large-time dynamics of Leb-a. e. point in the basin. [Wang-Young 2003]
Another version of the same idea: Generic supercritical Hopf bifurcations i. e. pair of cx eigenvalues cross Imaxis at = limit cycle, radius Normal form : generic : Introduce twist number Theorem. (2 D) If we apply with period T a kick such that is large enough , then for large enough T, a positive parameter set of flow-maps have an ergodic SRB measure that controls the large-time dynamics of Leb-a. e. point in the basin. [Wang-Young 2003]
Another version of the same idea: Generic supercritical Hopf bifurcations i. e. pair of cx eigenvalues cross Imaxis at = limit cycle, radius Normal form : generic : Introduce twist number Theorem. (2 D) If we apply with period T a kick such that is large enough , then for large enough T, a positive parameter set of flow-maps have an ergodic SRB measure that controls the large-time dynamics of Leb-a. e. point in the basin. [Wang-Young 2003]
Geometric meaning of Above works in any dim. Here is a version for evolutionary PDEs: THEOREM Hilbert space [Lu-Wang-Young 2013] Unforced system : assume generic supercrit Hopf bif at arbitrary, Forcing (last term) : smooth THEN for large enough, there is a pos Leb meas set for which the flow-map of measure. has an attractor w/ SRB Consequently, there is an open set in with pos Lyap exp ``a. e. ”
Geometric meaning of Above works in any dim. Here is a version for evolutionary PDEs: THEOREM Hilbert space [Lu-Wang-Young 2013] Unforced system : assume generic supercrit Hopf bif at arbitrary, Forcing (last term) : smooth THEN for large enough, there is a pos Leb meas set for which the flow-map of measure. has an attractor w/ SRB Consequently, there is an open set in with pos Lyap exp ``a. e. ”
Geometric meaning of Above works in any dim. Here is a version for evolutionary PDEs: THEOREM Hilbert space [Lu-Wang-Young 2013] Unforced system : assume generic supercrit Hopf bif at arbitrary, Forcing (last term) : smooth THEN for large enough, there is a pos Leb meas set for which the flow-map of measure. has an attractor w/ SRB Consequently, there is an open set in with pos Lyap exp ``a. e. ”
Geometric meaning of Above works in any dim. Here is a version for evolutionary PDEs: THEOREM Hilbert space [Lu-Wang-Young 2013] Unforced system : assume generic supercrit Hopf bif at arbitrary, Forcing (last term) : smooth THEN for large enough, there is a pos Leb meas set for which the flow-map of measure. has an attractor w/ SRB Consequently, there is an open set in with pos Lyap exp ``a. e. ”
Geometric meaning of Above works in any dim. Here is a version for evolutionary PDEs: THEOREM Hilbert space [Lu-Wang-Young 2013] Unforced system : assume generic supercrit Hopf bif at arbitrary, Forcing (last term) : smooth THEN for large enough, there is a pos Leb meas set for which the flow-map of measure. has an attractor w/ SRB Consequently, there is an open set in with pos Lyap exp ``a. e. ”
Recall: For dynamial systms in finite dim, an often adopted point of view is observable events = positive Leb measure sets In infinite dimensions: what plays the role of Leb measure ? More to the point, what is a ``typical” solution for a PDE ? One possible answer: finite-dimensional probes Say a Banach space property P holds ``almost everywhere” if for large classes (large in relevant sense) of embedded finite-dim disks D, P holds for Leb-a. e. point in D. Cf concept of “Haar null” [Lindenstrauss], prevalence/shyness [Hunt-Sauer-York Pictorial idea of proof for Hilbert sp result: has 2 D center mfld with attractor and codim 2 strong stable foliation absolutely continuous i. e. , Leb meas class preserved by holonomy property Leb-a. e. on passed to all 2 D mflds transversal to “a. e. ” pt in open set in Banach sp have pos Lyap e
Recall: For dynamial systms in finite dim, an often adopted point of view is observable events = positive Leb measure sets In infinite dimensions: what plays the role of Leb measure ? More to the point, what is a ``typical” solution for a PDE ? One possible answer: finite-dimensional probes Say a Banach space property P holds ``almost everywhere” if for large classes (large in relevant sense) of embedded finite-dim disks D, P holds for Leb-a. e. point in D. Cf concept of “Haar null” [Lindenstrauss], prevalence/shyness [Hunt-Sauer-York Pictorial idea of proof for Hilbert sp result: has 2 D center mfld with attractor and codim 2 strong stable foliation absolutely continuous i. e. , Leb meas class preserved by holonomy property Leb-a. e. on passed to all 2 D mflds transversal to “a. e. ” pt in open set in Banach sp have pos Lyap e
Recall: For dynamial systms in finite dim, an often adopted point of view is observable events = positive Leb measure sets In infinite dimensions: what plays the role of Leb measure ? More to the point, what is a ``typical” solution for a PDE ? One possible answer: finite-dimensional probes Say a Banach space property P holds ``almost everywhere” if for large classes (large in relevant sense) of embedded finite-dim disks D, P holds for Leb-a. e. point in D. Cf concept of “Haar null” [Lindenstrauss], prevalence/shyness [Hunt-Sauer-York Pictorial idea of proof for Hilbert sp result: has 2 D center mfld with attractor and codim 2 strong stable foliation absolutely continuous i. e. , Leb meas class preserved by holonomy property Leb-a. e. on passed to all 2 D mflds transversal to “a. e. ” pt in open set in Banach sp have pos Lyap e
Recall: For dynamial systms in finite dim, an often adopted point of view is observable events = positive Leb measure sets In infinite dimensions: what plays the role of Leb measure ? More to the point, what is a ``typical” solution for a PDE ? One possible answer: finite-dimensional probes Say a Banach space property P holds ``almost everywhere” if for large classes (large in relevant sense) of embedded finite-dim disks D, P holds for Leb-a. e. point in D. Cf concept of “Haar null” [Lindenstrauss], prevalence/shyness [Hunt-Sauer-York Pictorial idea of proof for Hilbert sp result: has 2 D center mfld with attractor and codim 2 strong stable foliation absolutely continuous i. e. , Leb meas class preserved by holonomy property Leb-a. e. on passed to all 2 D mflds transversal to “a. e. ” pt in open set in Banach sp have pos Lyap e
Recall: For dynamial systms in finite dim, an often adopted point of view is observable events = positive Leb measure sets In infinite dimensions: what plays the role of Leb measure ? More to the point, what is a ``typical” solution for a PDE ? One possible answer: finite-dimensional probes Say a Banach space property P holds ``almost everywhere” if for large classes (large in relevant sense) of embedded finite-dim disks D, P holds for Leb-a. e. point in D. Cf concept of “Haar null” [Lindenstrauss], prevalence/shyness [Hunt-Sauer-York Pictorial idea of proof for Hilbert sp result: has 2 D center mfld with attractor and codim 2 strong stable foliation absolutely continuous i. e. , Leb meas class preserved by holonomy property Leb-a. e. on passed to all 2 D mflds transversal to “a. e. ” pt in open set in Banach sp have pos Lyap e
Recall: For dynamial systms in finite dim, an often adopted point of view is observable events = positive Leb measure sets In infinite dimensions: what plays the role of Leb measure ? More to the point, what is a ``typical” solution for a PDE ? One possible answer: finite-dimensional probes Say a Banach space property P holds ``almost everywhere” if for large classes (large in relevant sense) of embedded finite-dim disks D, P holds for Leb-a. e. point in D. Cf concept of “Haar null” [Lindenstrauss], prevalence/shyness [Hunt-Sauer-York Pictorial idea of proof for Hilbert sp result: has 2 D center mfld with attractor and codim 2 strong stable foliation absolutely continuous i. e. , Leb meas class preserved by holonomy property Leb-a. e. on passed to all 2 D mflds transversal to “a. e. ” pt in open set in Banach sp have pos Lyap e
Recall: For dynamial systms in finite dim, an often adopted point of view is observable events = positive Leb measure sets In infinite dimensions: what plays the role of Leb measure ? More to the point, what is a ``typical” solution for a PDE ? One possible answer: finite-dimensional probes Say a Banach space property P holds ``almost everywhere” if for large classes (large in relevant sense) of embedded finite-dim disks D, P holds for Leb-a. e. point in D. Cf concept of “Haar null” [Lindenstrauss], prevalence/shyness [Hunt-Sauer-York Pictorial idea of proof for Hilbert sp result: has 2 D center mfld with attractor and codim 2 strong stable foliation absolutely continuous i. e. , Leb meas class preserved by holonomy property Leb-a. e. on passed to all 2 D mflds transversal to “a. e. ” pt in open set in Banach sp have pos Lyap e
A concrete example : periodically forced Brusselator (1 D) simplified model of autocatalytic chemical reaction with diffusion + periodic forcing [Lefevre-Progogine 1968] Dirichlet or Neumann boundary conditions constants, concentrations of certain initial substances concentrations of two intermediates Two other chemicals produced; they play no role. Unforced equation : For fixed is stationary solution supercritical Hopf bifurcation occurs at twist as Forcing (Neumann bc) : e. g. Theorem applies to give -- horseshoes for open sets of parameters (e. g. T) -- SRB meas and positive Lyap exp ``a. e. ” on an open set of function space ,
A concrete example : periodically forced Brusselator (1 D) simplified model of autocatalytic chemical reaction with diffusion + periodic forcing [Lefevre-Progogine 1968] Dirichlet or Neumann boundary conditions constants, concentrations of certain initial substances concentrations of two intermediates Two other chemicals produced; they play no role. Unforced equation : For fixed is stationary solution supercritical Hopf bifurcation occurs at twist as Forcing (Neumann bc) : e. g. Theorem applies to give -- horseshoes for open sets of parameters (e. g. T) -- SRB meas and positive Lyap exp ``a. e. ” on an open set of function space ,
A concrete example : periodically forced Brusselator (1 D) simplified model of autocatalytic chemical reaction with diffusion + periodic forcing [Lefevre-Progogine 1968] Dirichlet or Neumann boundary conditions constants, concentrations of certain initial substances concentrations of two intermediates Two other chemicals produced; they play no role. Unforced equation : For fixed is stationary solution supercritical Hopf bifurcation occurs at twist as Forcing (Neumann bc) : e. g. Theorem applies to give -- horseshoes for open sets of parameters (e. g. T) -- SRB meas and positive Lyap exp ``a. e. ” on an open set of function space ,
A concrete example : periodically forced Brusselator (1 D) simplified model of autocatalytic chemical reaction with diffusion + periodic forcing [Lefevre-Progogine 1968] Dirichlet or Neumann boundary conditions constants, concentrations of certain initial substances concentrations of two intermediates Two other chemicals produced; they play no role. Unforced equation : For fixed is stationary solution supercritical Hopf bifurcation occurs at twist as Forcing (Neumann bc) : e. g. Theorem applies to give -- horseshoes for open sets of parameters (e. g. T) -- SRB meas and positive Lyap exp ``a. e. ” on an open set of function space ,
A concrete example : periodically forced Brusselator (1 D) simplified model of autocatalytic chemical reaction with diffusion + periodic forcing [Lefevre-Progogine 1968] Dirichlet or Neumann boundary conditions constants, concentrations of certain initial substances concentrations of two intermediates Two other chemicals produced; they play no role. Unforced equation : For fixed is stationary solution supercritical Hopf bifurcation occurs at twist as Forcing (Neumann bc) : e. g. Theorem applies to give -- horseshoes for open sets of parameters (e. g. T) -- SRB meas and positive Lyap exp ``a. e. ” on an open set of function space ,
Recapitulating. . . 100 years ago, van der Pol observed “irregular noise” in periodically forced relaxation oscillator (analytically very difficult) Same spirit but simpler (single time-scale) : periodic kicking of limit cycles First example: 2 D linear shear flow Many dynamical types as shear varies -- low shear regimes: invariant curve (rotation vs sinks/saddles) -- increasing shear: inv curve breaks, horseshoes develop -- high-shear regimes: competition between horseshoes with sinks (open set) and observable chaos (pos meas set of parameters) Idea of observable events = positive Leb meas sets For dissipative systems (with attractors): SRB measures -- measure discovered for uniformly hyperbolic attractors -- without invariant cones: same theory, but existence not guaranteed Main examples of nonunif hyperbolic attractors shown to have SRB meas: Technically: “rank-one”, phenomenologically: shear-induced cha Examples are extensions of linear shear flow arbitrary limit cycles, Hopf bifurcations, any dim, including PDEs
Recapitulating. . . 100 years ago, van der Pol observed “irregular noise” in periodically forced relaxation oscillator (analytically very difficult) Same spirit but simpler (single time-scale) : periodic kicking of limit cycles First example: 2 D linear shear flow Many dynamical types as shear varies -- low shear regimes: invariant curve (rotation vs sinks/saddles) -- increasing shear: inv curve breaks, horseshoes develop -- high-shear regimes: competition between horseshoes with sinks (open set) and observable chaos (pos meas set of parameters) Idea of observable events = positive Leb meas sets For dissipative systems (with attractors): SRB measures -- measure discovered for uniformly hyperbolic attractors -- without invariant cones: same theory, but existence not guaranteed Main examples of nonunif hyperbolic attractors shown to have SRB meas: Technically: “rank-one”, phenomenologically: shear-induced cha Examples are extensions of linear shear flow arbitrary limit cycles, Hopf bifurcations, any dim, including PDEs
Recapitulating. . . 100 years ago, van der Pol observed “irregular noise” in periodically forced relaxation oscillator (analytically very difficult) Same spirit but simpler (single time-scale) : periodic kicking of limit cycles First example: 2 D linear shear flow Many dynamical types as shear varies -- low shear regimes: invariant curve (rotation vs sinks/saddles) -- increasing shear: inv curve breaks, horseshoes develop -- high-shear regimes: competition between horseshoes with sinks (open set) and observable chaos (pos meas set of parameters) Idea of observable events = positive Leb meas sets For dissipative systems (with attractors): SRB measures -- measure discovered for uniformly hyperbolic attractors -- without invariant cones: same theory, but existence not guaranteed Main examples of nonunif hyperbolic attractors shown to have SRB meas: Technically: “rank-one”, phenomenologically: shear-induced cha Examples are extensions of linear shear flow arbitrary limit cycles, Hopf bifurcations, any dim, including PDEs
Recapitulating. . . 100 years ago, van der Pol observed “irregular noise” in periodically forced relaxation oscillator (analytically very difficult) Same spirit but simpler (single time-scale) : periodic kicking of limit cycles First example: 2 D linear shear flow Many dynamical types as shear varies -- low shear regimes: invariant curve (rotation vs sinks/saddles) -- increasing shear: inv curve breaks, horseshoes develop -- high-shear regimes: competition between horseshoes with sinks (open set) and observable chaos (pos meas set of parameters) Idea of observable events = positive Leb meas sets For dissipative systems (with attractors): SRB measures -- measure discovered for uniformly hyperbolic attractors -- without invariant cones: same theory, but existence not guaranteed Main examples of nonunif hyperbolic attractors shown to have SRB meas: Technically: “rank-one”, phenomenologically: shear-induced cha Examples are extensions of linear shear flow arbitrary limit cycles, Hopf bifurcations, any dim, including PDEs
Recapitulating. . . 100 years ago, van der Pol observed “irregular noise” in periodically forced relaxation oscillator (analytically very difficult) Same spirit but simpler (single time-scale) : periodic kicking of limit cycles First example: 2 D linear shear flow Many dynamical types as shear varies -- low shear regimes: invariant curve (rotation vs sinks/saddles) -- increasing shear: inv curve breaks, horseshoes develop -- high-shear regimes: competition between horseshoes with sinks (open set) and observable chaos (pos meas set of parameters) Idea of observable events = positive Leb meas sets For dissipative systems (with attractors): SRB measures -- measure discovered for uniformly hyperbolic attractors -- without invariant cones: same theory, but existence not guaranteed Main examples of nonunif hyperbolic attractors shown to have SRB meas: Technically: “rank-one”, phenomenologically: shear-induced cha Examples are extensions of linear shear flow arbitrary limit cycles, Hopf bifurcations, any dim, including PDEs
Recapitulating. . . 100 years ago, van der Pol observed “irregular noise” in periodically forced relaxation oscillator (analytically very difficult) Same spirit but simpler (single time-scale) : periodic kicking of limit cycles First example: 2 D linear shear flow Many dynamical types as shear varies -- low shear regimes: invariant curve (rotation vs sinks/saddles) -- increasing shear: inv curve breaks, horseshoes develop -- high-shear regimes: competition between horseshoes with sinks (open set) and observable chaos (pos meas set of parameters) Idea of observable events = positive Leb meas sets For dissipative systems (with attractors): SRB measures -- measure discovered for uniformly hyperbolic attractors -- without invariant cones: same theory, but existence not guaranteed Main examples of nonunif hyperbolic attractors shown to have SRB meas: Technically: “rank-one”, phenomenologically: shear-induced cha Examples are extensions of linear shear flow arbitrary limit cycles, Hopf bifurcations, any dim, including PDEs
Recapitulating. . . 100 years ago, van der Pol observed “irregular noise” in periodically forced relaxation oscillator (analytically very difficult) Same spirit but simpler (single time-scale) : periodic kicking of limit cycles First example: 2 D linear shear flow Many dynamical types as shear varies -- low shear regimes: invariant curve (rotation vs sinks/saddles) -- increasing shear: inv curve breaks, horseshoes develop -- high-shear regimes: competition between horseshoes with sinks (open set) and observable chaos (pos meas set of parameters) Idea of observable events = positive Leb meas sets For dissipative systems (with attractors): SRB measures -- measure discovered for uniformly hyperbolic attractors -- without invariant cones: same theory, but existence not guaranteed Main examples of nonunif hyperbolic attractors shown to have SRB meas: Technically: “rank-one”, phenomenologically: shear-induced cha Examples are extensions of linear shear flow arbitrary limit cycles, Hopf bifurcations, any dim, including PDEs
Recapitulating. . . 100 years ago, van der Pol observed “irregular noise” in periodically forced relaxation oscillator (analytically very difficult) Same spirit but simpler (single time-scale) : periodic kicking of limit cycles First example: 2 D linear shear flow Many dynamical types as shear varies -- low shear regimes: invariant curve (rotation vs sinks/saddles) -- increasing shear: inv curve breaks, horseshoes develop -- high-shear regimes: competition between horseshoes with sinks (open set) and observable chaos (pos meas set of parameters) Idea of observable events = positive Leb meas sets For dissipative systems (with attractors): SRB measures -- measure discovered for uniformly hyperbolic attractors -- without invariant cones: same theory, but existence not guaranteed Main examples of nonunif hyperbolic attractors shown to have SRB meas: Technically: “rank-one”, phenomenologically: shear-induced cha Examples are extensions of linear shear flow arbitrary limit cycles, Hopf bifurcations, any dim, including PDEs
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