Control of chaos in Hamiltonian systems

Control of chaos in Hamiltonian systems G. Ciraolo, C. Chandre, R. Lima, M. Vittot Centre de Physique Théorique CNRS, Marseille M. Pettini Osservatorio Astrofisico di Arcetri, Università di Firenze Ph. Ghendrih Association Euratom-CEA, CEA Cadarache Contact: ciraolo@cpt.univ-mrs.fr Financial support: Association Euratom-CEA, CNRS, INFM References: J. Phys. A: Math. Gen. 37, 3589 (4) J. Phys. A: Math. Gen. 37, 6337 (4) Phys. Rev. E 69, 5613 (4) Nonlinearity 18 (5) Euratom Fréjus October, 4

Our strategy of control 1 original system dx (,) dt = X x t controlled system dx (,) c dt = X x t with X c = X + f chaotic dynamics regular dynamics control term Aim: find a small and apt control f > small modification of the system / great influence on the dynamics

Our strategy of control Aim: H = H +εv chaotic Hc = H + εv + f regular f = εv obvious and useless solution tailoring the control term f Requirements on f : small with respect to the perturbation εv > here, we require that f = O( ε ) localized in phase space > accessible region, fewer energy for the control with a certain shape > robustness other requirements?

Physical situations > electrostatic turbulence : E B drift motion y x ϕ θ V d x c c y dt y = E B= B B V x where E = V ( xy,, t) > magnetic field lines dψ H = dϕ θ dθ H =+ dϕ ψ H ( ψθ,, ϕ) = H ( ψ ) + εh ( ψ,, θ ϕ) 1 > atomic physics : atoms in external fields, traps > particle accelerators, free electron laser,...?

I. Global control of Hamiltonian systems : - method and illustration - application to edge plasma turbulence II. Localised control of Hamiltonian systems - method and illustration - application to magnetic field lines

In general Lie formalism In action-angle angle variables: H = H + V H( A, ϕ ) = H( A) + Vk( A) e k n where ( Αϕ, ) T n n i k ϕ LH given by LHV = { H, V} {, } = Poisson bracket L H V H V V H = A ϕ A ϕ n = iω k V k k ( A) e i k ϕ where H ω = A Γ pseudo inverse of L : L H Γ = H L H Γ V Vk( A) = n i ω k k ω k e i k ϕ R projector: R = 1 LH Γ RV V ( A) e = n k ω k= k i k ϕ

Global control Proposition 1 : H = H + V + f H + RV c and are canonically conjugate where f n ( 1) n = LΓ V ( nr + 1) V ( n + 1)! n= 1 V ε if is of order is of order f O ( ε ) > f = n= f n where f n is of order ε n H > If ω = is non-resonant and in many other situations, H + RV is integrable A

Control of chaotic transport in a model for E B drift in magnetized plasmas y B E( xyt,, ) B x V d x c c y where ( xy,, t) dt y = E B= E= V B B V x V is the Hamiltonian x, y are conjugate variables Algorithm for the computation of the control term f Numerical investigation: - of the effect of the control term (in practice f ) - of its robustness

Control of E B E B drift Electrostatic potential known on a spatio-temporal grid. Vxyt (,,) Vxyt= (,, ) 1) Expansion of iωk Vxyt (,,) Vk (, xy) e k t ) Computation of ΓV = ω k Vk (, x y) iωk e i ω k t 3) Computation of f 1 = Γ { V, V } where {, } = x y y x f( x,, y t = ) 4) Higher orders? ( recall: f = f n ). n= f n 1 = Γ n { V, f } n 1

Example: Vxyt= (,, ) ak Vxyt (,,) = sin ( π ( nx+ my) +ϕ 3/ nmk ωkt) ( n + m ) mnk,, Γ Vxyt (,, = ) ϕnmk random phases.1.1.1.1 f3( xyt=,, ) f (,, xyt= ) 3.1.1 4.1.1

The control term is a small modification of the potential Vxyt (,,) Vx (, y, t) + f (,, x y t)

Trajectories of particles > Poincaré surface of section a =.8 without control with control f > Projection on the x axis

Diffusion of test particles Diffusion coefficient without control D M 1 = lim xi( t) xi() t Mt i= 1 with control f > significant reduction of diffusion by f Probability Distribution Function of step sizes Step size distance between two successive sign reversals of the velocity without control with control f > the control quenches the large steps

Robustness of the control : a crucial requirement experimentally accessible H + ε V + f H + ε V + f domain of efficient control H + εv H + εv H + εv + f H + εv + f uncontrolled Hamiltonian controlled Hamiltonian, computed theoretically controlled Hamiltonian, implemented experimentally

Robustness 1 > Modification of the amplitude of the control term : f δ f a =.7 H c = H +ε V + δ f without control > optimal control for δ = 1 > increasing its amplitude does not improve control > decreasing the amplitude does still give a good control > reducing energy to control the system (δ =.5 reduction of 3%)

Robustness if the potential is known on a coarse grained grid: truncation of the Fourier series of the control term f a =.8 without control control f > efficient control with few Fourier modes: 1 modes reduction of 5% > simplification of the control term > reduction of the energy necessary for the control

I. Global control of Hamiltonian systems : - method and illustration - application to edge plasma turbulence II. Localised control of Hamiltonian systems - method and illustration - application to magnetic field lines

Localised control : channeling chaos by building barriers Proposition : There exists a control term f of order ε H c ( ϕ) ( A ϕh(, ϕ) ) ( A, ϕ) = H( A, ϕ) + f Ω + εγ has an invariant torus at A = εγ H(, ϕ). It is given by f ( ϕ) = H ( εγ H(, ϕ), ϕ). ϕ ϕ such that > expansion of H around A = A, eg.., A = : Ω( x) Remark: for d.o.f. barrier to diffusion for n d.o.f. effective barriers η η x Advantages: > Explicit expression for the control term: existence and regularity > Explicit expression for the invariant torus which has been created > Persistence of the created torus for arbitrarily large values of ε (provided f V )

Algorithm of construction of a localised control term p on an example: Hpxt (,,) = +εvxt (,) 1) translation of the momentum p by ω HpExt (,,, ) = ω p+ E+ε Vxt (, ) + ) computation of Γ V = ( Γ V, Γ V) 1 k1 Γ V ϕ = V k1, k ω k k 1 + k ikx e 1 + kt 1 ( ) (,, t) 3) computation of f = H εγ V, εγ V x ε f = ϕ x k1 k k ωk + k, 1 1 4) equation of the invariant torus p (, x t) = ω ε 1 x V kk 1 k1 ω k + k k, k 1 t kk e t ikx ( + kt) V 1 kk 1 e p ikx ( + kt) 1 A = (, pe) ϕ = (,) xt ω = ( ω,1)

Localised control: forced pendulum p Hc = +ε x x t ε cos x cos( x t) ( cos + cos( )) + Ω( p p (,) x t ) ω ω 1 without control ε =.5 with control Poincaré section 1 trajectory control 4% of V on 1% phase space control 4% of V on 1% phase space > uncontrolled case : no barrier for ε.8

Localised control: channelling chaos by building barriers Creation of barriers at different locations Poincaré section Poincaré section Creation of set of barriers M p Hc = +ε x i i x ( cos + cos( x t) ) + ε f(,) x t Ω( p p (,) t ) i= 1 > Trapping of particles ε =.5 Poincaré section

Localised control: magnetic field lines Magnetic field line dynamics in a toroidal geometry dϑ H =+ dϕ ψ dψ H = dϕ ϑ ϕ θ Nearly axisymmetric case: H( ψϑϕ,, ) = H( ψ ) + εh1( ψ, ϑ, ϕ) regular part: H ( ψ ) = dψ q( ψ) ε =.3 safety factor: 4 q( ψ ) = ψ ψ+ ψ ( )( ) magnetic perturbation: [ m m ϑ ] sin ( ) H m n m d m/ 1( ψϑϕ,, ) = ( 1) ψ cos( ϑ ϕ) m π( m m)

Localised control: magnetic field lines around ψ ψ : H ( ψϑϕ,, ) = H ( ψ ) + εh ( ψ, ϑ, ϕ) + ε f ( ϑ, ϕ) Ω( ψ ψ + εγ H ( ψ, θ, ϕ) ) c 1 θ 1 ε =.3 ε =.3 control 5% f V on 7% phase space ε =.3

Summary and outlook > Global control of Hamiltonian systems - application to electrostatic turbulence -robustness > Localised control of Hamiltonian systems - application to magnetic field lines - barrier to diffusion > control of chaos in area-preserving maps [cf. Poster] > application on a TWT [cf. Doveil s talk] > extension to more realistic models in fusion plasmas > other applications to atomic physics, particle accelerators, free electron laser..?

Robustness 3 > phase error in f ϕ = ϕ + γ ϕ a =.7 err nm nm nm without control control f > 5% reduction of the diffusion coefficient for 5 % error in the phases