Pseudo-Chaotic Orbits of Kicked Oscillators

Dynamical Chaos and Non-Equilibrium Statistical Mechanics: From Rigorous Results to Applications in Nano-Systems August, 006 Pseudo-Chaotic Orbits of Kicked Oscillators J. H. Lowenstein, New York University Outline 1. One-dimensional oscillators driven by impulsive kicks, periodic in x and t Resonant sinx) kick amplitude; chaos and pseudo-chaos in stochastic webs. Resonant sawtooth oscillator with quadratic irrational parameter Models with diffusive, super-diffusive, sub-diffusive, ballistic long-t behavior 3. Resonant sawtooth oscillator with cubic irrational parameter Infinitely many invariant components; multi-fractal structure; long-t behavior 4. Nonresonant sawtooth oscillator Periodic orbits and elliptical islands Pseudo-hyperbolic orbits Time-reversal symmetry; symmetric and asymmetric?) periodic orbits Invariant measure of aperiodic orbits positive?) Lifting to the plane: numerical explorations

Kicked Oscillator with Non-Resonant Kick Amplitude x ) K y lx - y ) mod 1) = x x ) W y = ) lx mod 1) - y x l = cosp r) r = irrational rotation number Example: l = 1/ Local Phase Portrait Periodic and quasi-periodic orbits Pseudo-hyperbolic orbits Invariant curves?) Measures

Partial Construction of the Discontinuity Set G l = 1/ + 80 itns. Generator G 0

Reversibility of the Local Map K ) x K = y ) lx - y ) mod 1 x Let G interchange x and y coordinates: x ) y G = y ) x, G = 1 x ) y G y ) x K G ly - x ) mod 1 ) y ) y ly - x ) mod 1 Hence K ly - ly - x) mod 1) mod1) x mod 1 ) x ) = = y y y KGKG = 1, i.e. GKG=K -1

Decomposition of K G = reflection about main diagonal=time-reversal map: GKG -1 = K -1 Let H = KG then H = KGKG = K K -1 = 1. Thus: K factorizes into a product of involutions: K = H G, G = H = 1 Symmetry lines FixG), FixH) are the sets of points left invariant by G and H, respectively. FixG) is the main diagonal y=x. FixH) l+ l+3 l+1 l+ l l+1 l-1 l -1/l -1/l 1/l 1/l 0 1/ 1 0 1/ 1 0 1/ 1 0 1/ 1

Symmetric Periodic Orbits A periodic orbit O is symmetric if GO = O, i.e. x,y) belongs to O if and only if y,x) belongs to O. If O is not symmetric, then GO is another periodic orbit obtained by reflection about the diagonal. Questions: For a given l, are there infinitely many symmetric periodic orbits? Are there any non-symmetric periodic orbits? Theorem A symmetric periodic orbit of odd period n+1 has one point z on FixG) and one point K n z) on FixH). A symmetric periodic orbit of even period n has points, z and K n z), on FixG) and none on FixH), or vice versa. Algorithm for collecting the symmetric periodic orbits in order of increasing period: apply K iteratively on the symmetry lines. Whenever a segment intersects x=0, it splits into segments, one shifted horizontally by +1. Whenever a segment intersects a symmetry line, the intersection point lies on a symmetric periodic orbit

Counting Symmetric Periodic Orbits Nt) Nt) ~ t 0.83 Nt) = # symmetric orbits with period < t l = -1/, -3/, 1/3 t

Elliptical Islands Every periodic orbit which avoids the discontinuity lines is stable : each point on the orbit lies at the center of an ellipse whose size is determined by tangency with a boundary of the square. Its area is 1 A = d p 4 - l d = minimum coordinate distance from a boundary What is the total area occupied by the elliptical islands of symmetric periodic orbits? all periodic orbits? Ashwin's conjecture: the area occupied by all elliptical islands is strictly less than the area of the square = 1).

Numerical Experiment: Total Area of Elliptical Islands F. Vivaldi and J.H.Lowenstein, Nonlinearity 19 006), 1069-1097 A = 0.909 l = 3/, centered origin 37886 orbits, island diameter e > 10 7 Best fit: Ae) = 0.904-0.095 e 0.155 A = 0.900 A search for non-symmtric orbits with islands of diameter e> 10 7 yielded no candidates. Supports Ashwin's conjecture.

Pseudo-Hyperbolic Orbits A pseudo-hyperbolic orbit contains a sequence of points which starts on the discontinuity generator G 0 and ends on G G 0. Such a sequence can be symmetric or non-symmetric. Every point of the sequence is a transverse intersection point for two discontinuity lines. Resemblance to hyperbolic points. Algorithm for collecting pseudo-hyperbolic sequences in order of increasing length: iterate K on G 0. When, on the nth step, a segment intersects the y axis, G G 0, the intersection point is the terminus of a pseudo-hyperbolic sequence of length n+1. Removing that point splits the segment into two segments, one of which is shifted horizontally by +1. Continue iterating K on the daughter segments.

Counting Pseudo-Hyperbolic Sequences l = 1/ = # pseudo-hyperbolic sequences of length < t Nt) ~ t 1.09 Nt) ~ t 0.9

Invariant Curves : Numerical Explorations 0.11 0.105 0.09 0.095 0.105 0.11 0.095 0.09 0.085 0.11 0.105 0.08 0.085 0.09 0.095 0.105 0.11 0.095 0.09 0.085 0.08 0.105 0.10 0.1015 0.101 0.1005 0.0990.0995 0.10050.1010.10150.100.105 0.08 0.101 0.10115 0.1011 0.10105 0.1008 0.1009 0.1011 0.101 0.1013 0.1014 0.10095 0.1009 0.10085 0.1008

Slice through the "Invariant-Curve" Ribbon 0.1014 0.1016 0.1006

Numerical Explorations: Global Orbits for l = 1/ Mystery orbit x,y)=.11,.11) Local orbit 100,000 itns. Movies: Global orbit 1,000,000 x 4 itns Global orbit 1,000,000 x 40 itns Local orbit zoom 400,000 itns

Invariant Curve in Ribbon, Lifted to the Plane x,y) = 0.101, 0.101) Local orbit 100,000 itns. Movie: Global orbit 10,000,000 x 4 itns.

Pseudo-Hyperbolic Orbit, Lifted to the Plane x,y) = 0/3, 0) Local orbit 100,000 itns. Movies: Global orbit billiard) Global orbit 100,000,000 itns.

Pseudo-Hyperbolic Orbit, Lifted to the Plane x,y) = 1499/171346,1499/3469) Local orbit 100,000 itns. Movies: Global orbit 5000 x 4 itns. Global orbit 5,000,000 x 4 itns. Global orbit 1.5 x 10 9 x 4 itns

Chaotic-Looking Orbit x,y) = 0.05, 0.05) Local orbit 50,000 itns. Movie: Global orbit, 5,0000,000 x 4 itns.