Pseudo-Chaotic Orbits of Kicked Oscillators

Transcription

1 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

2 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

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

4 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

5 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

6 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

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

8 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).

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

10 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.

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

12 Invariant Curves : Numerical Explorations

13 Slice through the "Invariant-Curve" Ribbon

14 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

15 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.

16 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.

17 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

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

19 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