Geometric Dynamic Days Heidelberg Germany Oct 27 28

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

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

Van der Pol observed “irregular noise” in equivalently Cartwright & Littlewood (1945) found 2 stable period orbits with diff periods Levinson (1949) modified to piecewise constant Their work inspired Smale’s horseshoe (1960 s) 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)

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

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

Trapping region and attracting set Since for large enough is a trapping region, and is a compact set that attracts all initial conditions. Next slides: I will fix , write (smooth) and focus on varying and. Partial dynamical picture only (i. e. dynamics of on subset of -spac References 1. Q Wang & L-S Young, From invariant curves to strange attractors, CMP (20 [detailed proofs] 2. K Lin & L-S Young, Dynamics of periodically-kicked oscillators, J Fixed Poin & Application (2010) [review article]

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

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

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

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

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

Comparison with breaking of invariant curves in KAM theory KAM : area-preserving, “bands” of invariant curves, much more unstabl Kicked limit cycles : contracting, single inv curve, quite stable Similarities in how inv curves get broken: rational rotation numbers (with smaller denominators) more vulnerable fixed pt fp fp 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 along invariant curve + shear + contraction folds

Formation of horseshoes Continuing to increase shear or decrease damping : Theorem. There exist with horseshoes s. t. and gradient dynamics has a horseshoe for all. invariant circle

Definition. diffeomorphism, Say is a horseshoe if (1) is uniformly hyperbolic, i. e. expanding , (2) compact invariant set contracting is topologically conjugate to a full shift More generally, is top conj to Significance of (1) and (2) : (2) (1) dynamical complexity (as in flipping a coin) robustness, persistence Theorem. Hyperbolic invariant sets that are maximal in some neighborhoo persist under perturbations, i. e. , if = hyp inv set for and is near , then has a hyp inv set conj to . Note horseshoes can coexist with sinks. Existence of horseshoes existence of chaotic behavior ; trajectories can tend to stable equilibrium for Leb-a. e. initial condition.

Horseshoes are often accompanied by sinks through the creation of homoclinic tangencies, i. e. at tangential intersections of and , saddle fixed or periodic point Newhouse’s phenomenon of infinitely many sinks A stronger notion of chaotic behavior : positivity of Lyapunov exponents on positive Leb meas sets i. e. almost everywhere wrt Leb meas, or at least on positive Leb meas set (write ) “ observable chaos “ Note existence of horseshoes alone does not imply observable chaos. Can have horseshoes. Leb-a. e. in the presence of

Theorem. There exist if is s. t. and , then there exists with s. t. for all , roughly periodic (with period of limit cycle) for which the following holds : For all , has Leb-a. e. Remark: This is much harder than the horseshoe result. Here’s how it looks in parameter space : horseshoes gradient dynamics Competing scenarios in this region: horseshoes + sinks vs observable chaos invariant circle Competing scenarios here: rigid rotation vs sinks/saddles as we vary for

Summary Lyapunov exponents as function of kick period Increasing or decreasing with fixed : from invariant circle with mix of rigid rotation and sinks/saddes to gradient-like systems i. e. saddles + sinks to horseshoes + sinks to mix of horseshoes + sinks and “observable chaos” (= pos Lyap exp Leb-a. e. ) Varying fixing