Geometric Dynamic Days

1. Heidelberg, Germany Oct 27-28, 2017 Geometric Dynamic Days Lecture 1 : Dynamics of periodically kicked oscillators Lai-Sang Young Courant Institute, NYU http://www.cims.nyu.edu/~lsy/

2. In 1920s, van der Pol modeled a vacuum tube triode circuit with Letting and He observed square-shaped wave periodic solutions which he called relaxation oscillations Then he added a periodic forcing and reported (1927) ``irregular noise”. slow or critical manifold Analysis of the unforced equation: slow-fast system get

3. In 1920s, van der Pol modeled a vacuum tube triode circuit with Letting and He observed square-shaped wave periodic solutions which he called relaxation oscillations Then he added a periodic forcing and reported (1927) ``irregular noise”. slow or critical manifold Analysis of the unforced equation: slow-fast system get

4. In 1920s, van der Pol modeled a vacuum tube triode circuit with Letting and He observed square-shaped wave periodic solutions which he called relaxation oscillations Then he added a periodic forcing and reported (1927) ``irregular noise”. slow or critical manifold Analysis of the unforced equation: slow-fast system get

5. In 1920s, van der Pol modeled a vacuum tube triode circuit with Letting and He observed square-shaped wave periodic solutions which he called relaxation oscillations Then he added a periodic forcing and reported (1927) ``irregular noise”. slow or critical manifold Analysis of the unforced equation: slow-fast system get

6. In 1920s, van der Pol modeled a vacuum tube triode circuit with Letting and He observed square-shaped wave periodic solutions which he called relaxation oscillations Then he added a periodic forcing and reported (1927) ``irregular noise”. slow or critical manifold Analysis of the unforced equation: slow-fast system get

7. Van der Pol observed “irregular noise” in Levinson (1949) modified to piecewise constant Cartwright & Littlewood (1945) found 2 stable period orbits with diff periods equivalently Their work inspired Smale’s horseshoe (1960s) Levi (1981) proved existence of horseshoe for equation with above modification + piecewise constant forcing Haiduc (2009) published computer-assisted proof for existence of horseshoes for original equations Rest of lecture: will discuss the more tractable problem of periodic “kicking” of limit cycles (single time-scale)

8. Van der Pol observed “irregular noise” in Levinson (1949) modified to piecewise constant Cartwright & Littlewood (1945) found 2 stable period orbits with diff periods equivalently Their work inspired Smale’s horseshoe (1960s) Levi (1981) proved existence of horseshoe for equation with above modification + piecewise constant forcing Haiduc (2009) published computer-assisted proof for existence of horseshoes for original equations Rest of lecture: will discuss the more tractable problem of periodic “kicking” of limit cycles (single time-scale)

9. Van der Pol observed “irregular noise” in Levinson (1949) modified to piecewise constant Cartwright & Littlewood (1945) found 2 stable period orbits with diff periods equivalently Their work inspired Smale’s horseshoe (1960s) Levi (1981) proved existence of horseshoe for equation with above modification + piecewise constant forcing Haiduc (2009) published computer-assisted proof for existence of horseshoes for original equations Rest of lecture: will discuss the more tractable problem of periodic “kicking” of limit cycles (single time-scale)

10. Van der Pol observed “irregular noise” in Levinson (1949) modified to piecewise constant Cartwright & Littlewood (1945) found 2 stable period orbits with diff periods equivalently Their work inspired Smale’s horseshoe (1960s) Levi (1981) proved existence of horseshoe for equation with above modification + piecewise constant forcing Haiduc (2009) published computer-assisted proof for existence of horseshoes for original equations Rest of lecture: will discuss the more tractable problem of periodic “kicking” of limit cycles (single time-scale)

11. Van der Pol observed “irregular noise” in Levinson (1949) modified to piecewise constant Cartwright & Littlewood (1945) found 2 stable period orbits with diff periods equivalently Their work inspired Smale’s horseshoe (1960s) Levi (1981) proved existence of horseshoe for equation with above modification + piecewise constant forcing Haiduc (2009) published computer-assisted proof for existence of horseshoes for original equations Rest of lecture: will discuss the more tractable problem of periodic “kicking” of limit cycles (single time-scale)

12. Van der Pol observed “irregular noise” in Levinson (1949) modified to piecewise constant Cartwright & Littlewood (1945) found 2 stable period orbits with diff periods equivalently Their work inspired Smale’s horseshoe (1960s) Levi (1981) proved existence of horseshoe for equation with above modification + piecewise constant forcing Haiduc (2009) published computer-assisted proof for existence of horseshoes for original equations Rest of lecture: will discuss the more tractable problem of periodic “kicking” of limit cycles (single time-scale)

13. Note is a hyperbolic (exponentially attracting) limit cycle OK also for continuous forcing as concentrated on short time interval where i.e. system is given an impulsive force (or kick) Let = flow corresp to unforced equation The time- map of the kicked system is at times Periodic kicking of linear shear flow Unforced system : Kicked system :

14. Note is a hyperbolic (exponentially attracting) limit cycle OK also for continuous forcing as concentrated on short time interval where i.e. system is given an impulsive force (or kick) Let = flow corresp to unforced equation The time- map of the kicked system is at times Periodic kicking of linear shear flow Unforced system : Kicked system :

15. Note is a hyperbolic (exponentially attracting) limit cycle OK also for continuous forcing as concentrated on short time interval where i.e. system is given an impulsive force (or kick) Let = flow corresp to unforced equation The time- map of the kicked system is at times Periodic kicking of linear shear flow Unforced system : Kicked system :

16. Note is a hyperbolic (exponentially attracting) limit cycle OK also for continuous forcing as concentrated on short time interval where i.e. system is given an impulsive force (or kick) Let = flow corresp to unforced equation The time- map of the kicked system is at times Periodic kicking of linear shear flow Unforced system : Kicked system :

17. (note period of = 1) Observe similar progression if we fix and and let Fix for definiteness. or if we fix and and let = amount of shear = damping, or rate of contraction = amplitude of kick = time interval between kicks Geometry of = limit cycle e.g.

18. (note period of = 1) Observe similar progression if we fix and and let Fix for definiteness. or if we fix and and let = amount of shear = damping, or rate of contraction = amplitude of kick = time interval between kicks Geometry of = limit cycle e.g.

19. (note period of = 1) Observe similar progression if we fix and and let Fix for definiteness. or if we fix and and let = amount of shear = damping, or rate of contraction = amplitude of kick = time interval between kicks Geometry of = limit cycle e.g.

20. (note period of = 1) Observe similar progression if we fix and and let Fix for definiteness. or if we fix and and let = amount of shear = damping, or rate of contraction = amplitude of kick = time interval between kicks Geometry of = limit cycle e.g.

21. I will fix , write (smooth) and focus on varying and . Partial dynamical picture only (i.e. dynamics of on subset of -space) Next slides: for large enough is a trapping region, Trapping region and attracting set Since and is a compact set that attracts all initial conditions. References 1. Q Wang & L-S Young, From invariant curves to strange attractors, CMP (2002) [detailed proofs] 2. K Lin & L-S Young, Dynamics of periodically-kicked oscillators, J Fixed Point Theory & Application (2010) [review article]

22. I will fix , write (smooth) and focus on varying and . Partial dynamical picture only (i.e. dynamics of on subset of -space) Next slides: for large enough is a trapping region, Trapping region and attracting set Since and is a compact set that attracts all initial conditions. References 1. Q Wang & L-S Young, From invariant curves to strange attractors, CMP (2002) [detailed proofs] 2. K Lin & L-S Young, Dynamics of periodically-kicked oscillators, J Fixed Point Theory & Application (2010) [review article]

23. I will fix , write (smooth) and focus on varying and . Partial dynamical picture only (i.e. dynamics of on subset of -space) Next slides: for large enough is a trapping region, Trapping region and attracting set Since and is a compact set that attracts all initial conditions. References 1. Q Wang & L-S Young, From invariant curves to strange attractors, CMP (2002) [detailed proofs] 2. K Lin & L-S Young, Dynamics of periodically-kicked oscillators, J Fixed Point Theory & Application (2010) [review article]

24. I will fix , write (smooth) and focus on varying and . Partial dynamical picture only (i.e. dynamics of on subset of -space) Next slides: for large enough is a trapping region, Trapping region and attracting set Since and is a compact set that attracts all initial conditions. References 1. Q Wang & L-S Young, From invariant curves to strange attractors, CMP (2002) [detailed proofs] 2. K Lin & L-S Young, Dynamics of periodically-kicked oscillators, J Fixed Point Theory & Application (2010) [review article]

25. Theorem. Assume (depending on ) and Then is a simple closed curve to which all orbits of converge; is the graph of a function with small norm. Note for unforced flow , the strong stable manifold at Large enough invariant cones preserved by . Standard proof of center manifolds theorem applies. Q.E.D. For each , let , Add kick: -foliation on all of , in fact all of . viewed as , a circle diffeomorphism. Strong damping (or weak shear) regimes is a straight line of slope (more generally )

26. Theorem. Assume (depending on ) and Then is a simple closed curve to which all orbits of converge; is the graph of a function with small norm. Note for unforced flow , the strong stable manifold at Large enough invariant cones preserved by . Standard proof of center manifolds theorem applies. Q.E.D. For each , let , Add kick: -foliation on all of , in fact all of . viewed as , a circle diffeomorphism. Strong damping (or weak shear) regimes is a straight line of slope (more generally )

27. Theorem. Assume (depending on ) and Then is a simple closed curve to which all orbits of converge; is the graph of a function with small norm. Note for unforced flow , the strong stable manifold at Large enough invariant cones preserved by . Standard proof of center manifolds theorem applies. Q.E.D. For each , let , Add kick: -foliation on all of , in fact all of . viewed as , a circle diffeomorphism. Strong damping (or weak shear) regimes is a straight line of slope (more generally )

28. Theorem. Assume (depending on ) and Then is a simple closed curve to which all orbits of converge; is the graph of a function with small norm. Note for unforced flow , the strong stable manifold at Large enough invariant cones preserved by . Standard proof of center manifolds theorem applies. Q.E.D. For each , let , Add kick: -foliation on all of , in fact all of . viewed as , a circle diffeomorphism. Strong damping (or weak shear) regimes is a straight line of slope (more generally )

29. Theorem. Assume (depending on ) and Then is a simple closed curve to which all orbits of converge; is the graph of a function with small norm. Note for unforced flow , the strong stable manifold at Large enough invariant cones preserved by . Standard proof of center manifolds theorem applies. Q.E.D. For each , let , Add kick: -foliation on all of , in fact all of . viewed as , a circle diffeomorphism. Strong damping (or weak shear) regimes is a straight line of slope (more generally )

30. Let be an orientation-preserving homeomorphism, and and let be its lift The rotation number 1. relatively prime implies the existence of periodic orbits of period 2. and implies is topologically conjugate Fact: is a continuous and monotonically increasing; Theorem (Herman 70s): If is and for some small (same for all ) is topologically conjugate to rigid rotation by Then has positive Lebesgue measure. it is a devil’s staircase (constant on dense collection of intervals) Dynamics of circle diffeomorphisms (review) Elementary facts One-parameter family of circle diffeos (Arnold’s example)

31. Let be an orientation-preserving homeomorphism, and and let be its lift The rotation number 1. relatively prime implies the existence of periodic orbits of period 2. and implies is topologically conjugate Fact: is a continuous and monotonically increasing; Theorem (Herman 70s): If is and for some small (same for all ) is topologically conjugate to rigid rotation by Then has positive Lebesgue measure. it is a devil’s staircase (constant on dense collection of intervals) Dynamics of circle diffeomorphisms (review) Elementary facts One-parameter family of circle diffeos (Arnold’s example)

32. Let be an orientation-preserving homeomorphism, and and let be its lift The rotation number 1. relatively prime implies the existence of periodic orbits of period 2. and implies is topologically conjugate Fact: is a continuous and monotonically increasing; Theorem (Herman 70s): If is and for some small (same for all ) is topologically conjugate to rigid rotation by Then has positive Lebesgue measure. it is a devil’s staircase (constant on dense collection of intervals) Dynamics of circle diffeomorphisms (review) Elementary facts One-parameter family of circle diffeos (Arnold’s example)

33. Let be an orientation-preserving homeomorphism, and and let be its lift The rotation number 1. relatively prime implies the existence of periodic orbits of period 2. and implies is topologically conjugate Fact: is a continuous and monotonically increasing; Theorem (Herman 70s): If is and for some small (same for all ) is topologically conjugate to rigid rotation by Then has positive Lebesgue measure. it is a devil’s staircase (constant on dense collection of intervals) Dynamics of circle diffeomorphisms (review) Elementary facts One-parameter family of circle diffeos (Arnold’s example)

34. Let be an orientation-preserving homeomorphism, and and let be its lift The rotation number 1. relatively prime implies the existence of periodic orbits of period 2. and implies is topologically conjugate Fact: is a continuous and monotonically increasing; Theorem (Herman 70s): If is and for some small (same for all ) is topologically conjugate to rigid rotation by Then has positive Lebesgue measure. it is a devil’s staircase (constant on dense collection of intervals) Dynamics of circle diffeomorphisms (review) Elementary facts One-parameter family of circle diffeos (Arnold’s example)

35. Let be an orientation-preserving homeomorphism, and and let be its lift The rotation number 1. relatively prime implies the existence of periodic orbits of period 2. and implies is topologically conjugate Fact: is a continuous and monotonically increasing; Theorem (Herman 70s): If is and for some small (same for all ) is topologically conjugate to rigid rotation by Then has positive Lebesgue measure. it is a devil’s staircase (constant on dense collection of intervals) Dynamics of circle diffeomorphisms (review) Elementary facts One-parameter family of circle diffeos (Arnold’s example)

36. Dynamics of on invariant curve Write where For (e.g. ) , Then above converges to either topologically conj to irrational rotation for some , given by Conclude (depending on ) attracting set = invariant circle for all Arnold’s 1-parameter family and for , or (typically) finite # periodic sinks/sources alternating Singular limit as for fixed : Recall

37. Dynamics of on invariant curve Write where For (e.g. ) , Then above converges to either topologically conj to irrational rotation for some , given by Conclude (depending on ) attracting set = invariant circle for all Arnold’s 1-parameter family and for , or (typically) finite # periodic sinks/sources alternating Singular limit as for fixed : Recall

38. Dynamics of on invariant curve Write where For (e.g. ) , Then above converges to either topologically conj to irrational rotation for some , given by Conclude (depending on ) attracting set = invariant circle for all Arnold’s 1-parameter family and for , or (typically) finite # periodic sinks/sources alternating Singular limit as for fixed : Recall

39. Dynamics of on invariant curve Write where For (e.g. ) , Then above converges to either topologically conj to irrational rotation for some , given by Conclude (depending on ) attracting set = invariant circle for all Arnold’s 1-parameter family and for , or (typically) finite # periodic sinks/sources alternating Singular limit as for fixed : Recall

40. Dynamics of on : Breaking of invariant curves Then letting picture becomes For positive meas set of : conj to rotation Assume for definiteness “Typical picture” for open & dense set of : finite # sinks & sources alternating looks like = simple closed curve normally hyperbolic(i.e. normal contraction stronger than along curve)

41. Dynamics of on : Breaking of invariant curves Then letting picture becomes For positive meas set of : conj to rotation Assume for definiteness “Typical picture” for open & dense set of : finite # sinks & sources alternating looks like = simple closed curve normally hyperbolic(i.e. normal contraction stronger than along curve)

42. Dynamics of on : Breaking of invariant curves Then letting picture becomes For positive meas set of : conj to rotation Assume for definiteness “Typical picture” for open & dense set of : finite # sinks & sources alternating looks like = simple closed curve normally hyperbolic(i.e. normal contraction stronger than along curve)

43. Dynamics of on : Breaking of invariant curves Then letting picture becomes For positive meas set of : conj to rotation Assume for definiteness “Typical picture” for open & dense set of : finite # sinks & sources alternating looks like = simple closed curve normally hyperbolic(i.e. normal contraction stronger than along curve)

44. Then letting picture becomes Claim : when 1D map loses invertibility, inv curve breaks for with Sketch of proof: If = simple closed curve, and if then nhhd of in s.t. meets same -leaf 3 times and on attractive fixed pt in 2D , saddle fixed point in 2D Assume contradicting = simple closed curve ! This implies is foliated by of . Likewise

45. Then letting picture becomes Claim : when 1D map loses invertibility, inv curve breaks for with Sketch of proof: If = simple closed curve, and if then nhhd of in s.t. meets same -leaf 3 times and on attractive fixed pt in 2D , saddle fixed point in 2D Assume contradicting = simple closed curve ! This implies is foliated by of . Likewise

46. Then letting picture becomes Claim : when 1D map loses invertibility, inv curve breaks for with Sketch of proof: If = simple closed curve, and if then nhhd of in s.t. meets same -leaf 3 times and on attractive fixed pt in 2D , saddle fixed point in 2D Assume contradicting = simple closed curve ! This implies is foliated by of . Likewise

47. Then letting picture becomes Claim : when 1D map loses invertibility, inv curve breaks for with Sketch of proof: If = simple closed curve, and if then nhhd of in s.t. meets same -leaf 3 times and on attractive fixed pt in 2D , saddle fixed point in 2D Assume contradicting = simple closed curve ! This implies is foliated by of . Likewise

48. Then letting picture becomes Claim : when 1D map loses invertibility, inv curve breaks for with Sketch of proof: If = simple closed curve, and if then nhhd of in s.t. meets same -leaf 3 times and on attractive fixed pt in 2D , saddle fixed point in 2D Assume contradicting = simple closed curve ! This implies is foliated by of . Likewise

49. Similarities in how inv curves get broken: Diophantine rotation no = uniform distr or orbit on circle similar amount of and For kicked limit cycles, variation in net amount of and per kick period KAM : area-preserving, “bands” of invariant curves, much more unstable rational rotation numbers (with smaller denominators) more vulnerable Kicked limit cycles : contracting, single inv curve, quite stable along invariant curve + shear + contraction folds fp fixed pt fp Comparison with breaking of invariant curves in KAM theory

50. Similarities in how inv curves get broken: Diophantine rotation no = uniform distr or orbit on circle similar amount of and For kicked limit cycles, variation in net amount of and per kick period KAM : area-preserving, “bands” of invariant curves, much more unstable rational rotation numbers (with smaller denominators) more vulnerable Kicked limit cycles : contracting, single inv curve, quite stable along invariant curve + shear + contraction folds fp fixed pt fp Comparison with breaking of invariant curves in KAM theory