Analytical and Numerical Methods for Nonlinear Dynamics Tim Zolkin 1 - PowerPoint PPT Presentation

Analytical and Numerical Methods for Nonlinear Dynamics Tim Zolkin 1 1 Fermi National Accelerator Laboratory July 24, 2019 Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

0 . INTRODUCTION Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

Basic definitions Consider a mapping ( map ) T : M → M defined by a function f ζ n +1 = f ( ζ n ) , ζ i ∈ M . Manifold M can be R n , C n , S n , T n , etc.. The trajectory of ζ 0 is the finite set � � ζ 0 , T ( ζ 0 ) , T 2 ( ζ 0 ) , . . . , T n ( ζ 0 ) The orbit of ζ 0 , is a set of all points that can be reached � � . . . , T − 2 ( ζ 0 ) , T − 1 ( ζ 0 ) , ζ 0 , T ( ζ 0 ) , T 2 ( ζ 0 ) , . . . The n -cycle (or periodic orbit of period n ) is a solution of T n ( ζ 0 ) = ζ 0 Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

Symplectic mappings of the plane We will consider area-preserving mappings of the plane � ∂ q ′ /∂ q � q ′ = q ′ ( q , p ) , ∂ q ′ /∂ p det = 1 . p ′ = p ′ ( q , p ) , ∂ p ′ /∂ q ∂ p ′ /∂ p Reflection ∗ , ∗∗ , Ref Identity, Id Rotation, Rot � 1 � � cos θ � � cos 2 θ � − sin θ sin 2 θ 0 0 1 sin θ cos θ sin 2 θ − cos 2 θ Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

Integrable systems A map T in the plane is called integrable , if there exists a non- constant real valued continuous functions K ( q , p ), called integral , which is invariant under T : K ( q , p ) = K ( q ′ , p ′ ) ∀ ( q , p ) : where primes denote the application of the map, ( q ′ , p ′ ) = T ( q , p ). Example: Rotation transformation q ′ = q cos θ − p sin θ Rot ( θ ) : p ′ = q sin θ + p cos θ has the integral K ( q , p ) = q 2 + p 2 . Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

McMillan form of the map McMillan considered a special form of the map q ′ = p , M : p ′ = − q + f ( p ) , where f ( p ) is called force function (or simply force ). a. Fixed point p = q ∩ p = 1 2 f ( q ) . b. 2-cycles q = 1 2 f ( p ) ∩ p = 1 2 f ( q ) . Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

1D accelerator lattice with thin nonlinear lens, T = F ◦ M � y � ′ � cos Φ + α sin Φ � � y � β sin Φ M : = , ˙ − γ sin Φ cos Φ − α sin Φ ˙ y y � y � ′ � y � � 0 � F : = + , ˙ ˙ F ( y ) y y where α , β and γ are Courant-Snyder parameters at the thin lens location, and, Φ is the betatron phase advance of one period. Mapping in McMillan form after CT to ( q , p ), T = � F ◦ Rot ( − π/ 2) q = y , p = y (cos Φ + α sin Φ) + ˙ y β sin Φ , � F ( q ) = 2 q cos Φ + β F ( q ) sin Φ . Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

Example 1: Standard map/Chirikov-Taylor map/Chirikov standard map ( f = cos p ) ∆ E n +1 = ∆ E n + e V (sin φ n − sin φ s ) φ n +1 = φ n + 2 π h η β 2 E ∆ E n +1 Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

enon quadratic map ( f = p 2 ) Example 2: H´ Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

Turaev theorem Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

1 . PERTURBATION THEORY Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

Consider a map in McMillan form : q ′ = p , T : p ′ = − q + f ( p ) , where function f ( p ) is of the class C ∞ and will be referred to as a force function , or simply force . In order to construct a perturbation theory, we shall introduce a small positive parameter ǫ characterizing the amplitude of oscillations. It can be done using a change of variables ( q , p ) → ǫ ( q , p ): q ′ = p T : p ′ = − q + 1 2! p 2 + ǫ 2 c 3! p 3 + . . . . ǫ f ( ǫ p ) = − q + a p + ǫ b where we expanded the force function in a power series of ( ǫ p ) and b ≡ ∂ 2 c ≡ ∂ 3 a ≡ ∂ p f (0) , p f (0) , p f (0) , . . . . Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

Linearization of map q ′ = p T : p ′ = − q + a p + ǫ b 2! p 2 + ǫ 2 c 3! p 3 + . . . . Jacobian of transformation � ∂ q ′ � � 0 � ∂ q ′ 1 ∂ q ∂ p J T = = ∂ p ′ ∂ p ′ − 1 a ∂ q ∂ p Courant-Snyder invariant C . S . = p 2 − a p q + q 2 Betatron frequency µ = 1 2 π arccos a 2 Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

K ( n ) ( p ′ , q ′ ) − K ( n ) ( p , q ) = O ( ǫ n +1 ) We seek for an invariant of motion expanded in powers of a small parameter: K ( n ) = K 0 + ǫ K 1 + ǫ 2 K 2 + . . . + ǫ n K n such that K m are degree ( m + 2) polynomials K 0 = C 2 , 0 p 2 + C 1 , 1 p q + C 0 , 2 q 2 , K 1 = C 3 , 0 p 3 + C 2 , 1 p 2 q + C 1 , 2 p q 2 + C 0 , 3 q 3 , K 2 = C 4 , 0 p 4 + C 3 , 1 p 3 q + C 2 , 2 p 2 q 2 + C 1 , 3 p q 3 + C 0 , 4 q 4 , · · · . Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

Due to the first symmetry, K ( q , p ) = K ( p , q ), it is convenient to introduce the following notations: C . S . = Σ 2 − (2+ a ) Π = p 2 − a p q + q 2 Σ = p + q Π = p q Then we perform the expansion for even and odd orders of PT as K 0 = C . S . K 1 = A (1) ΠΣ 1 Π 2 + C (2) C . S . 2 K 2 = A (2) 1 K 3 = A (3) Π 2 Σ + A (3) ΠΣ C . S . 1 2 Π 3 + A (4) Π 2 C . S . + C (4) C . S . 3 K 4 = A (4) 1 2 K 5 = A (5) Π 3 Σ + A (5) Π 2 Σ C . S . + A (5) ΠΣ C . S . 2 1 2 3 . . . Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

Averaging 1. Canonical change of variables to Floquet coordinates � 1 / 4 √ � − 1 / 4 √ � � 1 − a 2 / 4 2 J cos( ϕ ) + a 1 − a 2 / 4 q = 2 J sin( ϕ ) , 2 � − 1 / 4 √ � 1 − a 2 / 4 p = 2 J sin( ϕ ) , 2. Rewriting the residual in terms of ( J , ϕ ) It is periodic function of ϕ , so its average over a full period vanishes: � 2 π � � K (2) ( q ′ , p ′ ) − K (2) ( q , p ) d ϕ = 0 . 0 3. Minimization of the average of the squared residual � 2 π � � 2 K (2) ( q ′ , p ′ ) − K (2) ( q , p ) I 1 = d ϕ 0 d and solve for C 1 from d C 1 I 1 = 0 Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

Approximated invariant for H´ enon map � r 3 r 4 Π 2 + C 1 C . S . 2 � K (3) r 3 Σ Π ǫ 1 + b 2 sex = C . S . − b ǫ 2 − � � � � r 4 r 5 Σ Π 2 − b 2 b 2 − b ǫ 3 r 3 r 4 r 5 − 2 C 1 Σ Π C . S . r 3 �K (0) sex � = r 1 r 2 C . S . �K (1) sex � = r 1 r 2 r 3 C . S . − r 1 r 2 Σ Π ǫ b � 4 C . S . 2 � r 1 r 2 Π 2 + 5 �K (2) ǫ 2 b 2 sex � = r 1 r 2 r 3 r 4 C . S . − r 1 r 2 r 4 Σ Π ǫ b + �K (3) sex � = r 1 r 2 r 3 r 4 r 5 C . S . − r 1 r 2 r 4 r 5 Σ Π ǫ b + � P 0 C . S . 2 � � � r 2 r 5 Π 2 + P 1 ǫ 2 b 2 − r 1 r 2 Σ Π 2 + 7 P 2 ǫ 3 b 3 + r 1 P 0 Σ Π C . S . where r 1 = a − 2 r 2 = a + 2 r 3 = a + 1 √ √ r 5 = ( a + 1+ 5 )( a + 1 − 5 r 4 = a ) 2 2 Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

a. Resonance cases (Sextupole on a 1/4 resonance) Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

b. Islands (Octupole below 1/4 resonance) Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

c. Unstable fixed point (Octupole below 1/2 resonance) Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

d. Frequency as a function of amplitude (Octupole above 1/4 resonance) Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

2 . DELIVERY RING EXTRACTION FOR Mu2e Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

Implementation of Resonant Extraction in the Delivery Ring for Mu2e Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

Steps: 0. Prepare a map in polynomial form q ′ = ( a p 2 + b p q + c q 2 ) + ( d p 3 + e p 2 q + f p q 2 + g q 3 ) ǫ + . . . p ′ = (¯ a p 2 + ¯ d p 3 + ¯ f p q 2 + ¯ c q 2 ) + ( ¯ e p 2 q + ¯ g q 3 ) ǫ + . . . b p q + ¯ 1. Define a Residue function K ( n ) ( p ′ , q ′ ) − K ( n ) ( p , q ) = Res ( K ( n ) , k ) + O ( ǫ k +1 ) Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

2. Seek for an invariant of motion expanded in powers of ǫ K ( n ) = K 0 + ǫ K 1 + ǫ 2 K 2 + . . . + ǫ n K n such that K m are degree ( m + 2) polynomials K 0 = C 2 , 0 p 2 + C 1 , 1 p q + C 0 , 2 q 2 , K 1 = C 3 , 0 p 3 + C 2 , 1 p 2 q + C 1 , 2 p q 2 + C 0 , 3 q 3 , C 4 , 0 p 4 + C 3 , 1 p 3 q + C 2 , 2 p 2 q 2 + C 1 , 3 p q 3 + C 0 , 4 q 4 ❍❍❍ ✟ K 2 = ✟✟✟ ❍ + C 1 K 2 0 K 3 = C 5 , 0 p 5 + C 4 , 1 p 4 q + C 3 , 2 p 3 q 2 + C 2 , 3 p 2 q 3 + C 1 , 4 p q 4 + C 0 , 5 q 5 C 6 , 0 p 6 + C 5 , 1 p 5 q + C 4 , 2 p 4 q 2 + C 3 , 3 p 3 q 3 + C 2 , 4 p 2 q 4 ❍❍❍ ✟ K 4 = ✟✟✟ ❍ + C 1 , 5 p q 5 + C 0 , 6 q 6 + C 2 K 3 0 · · · Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics

2.1 Determine C i , j ( C i ) Res ( K ( n ) , n ) = 0 2.2 Determine C i using averaging procedure � 2 π d I Res 2 ( K ( n ) , n + 1) d φ = 0 I = where d C i 0 and for K (0) = b p 2 + ( a − d ) p q − c q 2 we use √ 2 b cos φ √ q = b c − ( a − d ) 2 � � ( a − d ) cos φ p = − 1 √ sin φ + √ b c − ( a − d ) 2 b 2.3 If using parameters, κ i , remove resonances �K� = Pol ( q , p ) → f ( κ i ) × �K� f ( κ i ) Tim Zolkin Analytical and Numerical Methods for Nonlinear Dynamics