new perspectives into the chaotic dynamics in molecules · 2020. 5. 22. · fabio revuelta....
TRANSCRIPT
Fabio Revuelta
Dynamics Days Central Asia VI 5 June 2020
NEW PERSPECTIVES INTO THE CHAOTICDYNAMICS IN MOLECULES
Grupo de Sistemas ComplejosUniversidad Politécnica de Madrid (Spain)
NOLINEAL 20-21
12th International Conference onNonlinear Mathematics and Physics
30 June – 2 July 2021
Madrid (Spain)
https://www.gsc.upm.es/nolineal2020
NOLINEAL 20-21
12th International Conference onNonlinear Mathematics and Physics
30 June – 2 July 2021
Madrid (Spain)
https://www.gsc.upm.es/nolineal2020
OUTLINE
IntroductionSystemMethodologyResultsConclusions
CHAOS INDICATORS
PSOS - Poincaré Surface of section
Lyapunov exponent
Frequency Analisys - Laskar
SALI and GALI (Small and Generalized ALignment Index - Skokos.
FLI (Fast Lyapunov Indicator) - Lega, Guzzo, Froeschlé.
OFLI (Orthogonal Fast Lyapunov Indicator) - Barrio.
MEGNO (Mean Exponential Growth Factor of Nearby Orbits) - Cincotta, Simó
…
ESSENTIALS
ESSENTIALS
ESSENTIALSReference trajectory
ESSENTIALSReference trajectory
Nearbytrajectory 1
ESSENTIALSReference trajectory
Nearbytrajectory 1
ESSENTIALSReference trajectory
Nearbytrajectory 1
Nearbytrajectory 2
ESSENTIALSReference trajectory
Nearbytrajectory 1
Nearbytrajectory 2
ESSENTIALSReference trajectory
Nearbytrajectory 1
Nearbytrajectory 2
ESSENTIALSReference trajectory
Nearbytrajectory 1
Nearbytrajectory 2
ESSENTIALSReference trajectory
Nearbytrajectory 1
Nearbytrajectory 2
ESSENTIALSReference trajectory
ESSENTIALSReference trajectory
Lagrangian descriptors: allthe information encoded in the trajectory itself!
SYSTEM
WHY MOLECULES???
1. Atomic and molecular systems: excellentplatforms for application of dynamicalsystems theory
2. ⇑⇑⇑ anharmonic3. “Simple” (2 dof)
SYSTEM
N
Li
C
SYSTEM
N
Li
C
SYSTEM
N
Li
C
SYSTEM
N
Li
C
SYSTEM
N
Li
C
SYSTEM
N
Li
C
SYSTEM
1. 3 atoms: 9 dof2. No rotation: - 6 dof3. Coordinates: R, θ, r NC
Li
SYSTEM
N
Li
C
SYSTEM
N
RLi
C
SYSTEM
N
RθLi
C
SYSTEM
N
RθLi
Cr
SYSTEM
N
RθLi
Cr≃req
SYSTEM
N
RθLi
Cr≃req
(2 DOF)
HAMILTONIAN
HAMILTONIAN
HAMILTONIAN
HAMILTONIAN
HAMILTONIAN
HAMILTONIAN
HAMILTONIAN
HAMILTONIAN
POTENTIAL ENERGY SURFACENC
Li Rθ
POTENTIAL ENERGY SURFACENC
Li Rθ
POTENTIAL ENERGY SURFACENC
Li Rθ
Absoluteminimum
POTENTIAL ENERGY SURFACENC
Li Rθ
LiC N
Absoluteminimum
POTENTIAL ENERGY SURFACENC
Li Rθ
LiC N
Relativeminimum Absolute
minimum
POTENTIAL ENERGY SURFACENC
Li Rθ
NLi C LiC N
Relativeminimum Absolute
minimum
POTENTIAL ENERGY SURFACENC
Li Rθ
NLi C LiC N
Relativeminimum Absolute
minimumMEP
POTENTIAL ENERGY SURFACENC
Li Rθ
NLi C LiC N
Relativeminimum Absolute
minimumSaddle
MEP
POTENTIAL ENERGY SURFACENC
Li
r
Methodology
LAGRANGIAN DESCRIPTORS
Figure: C. Mendoza, A. M. Mancho,Nonlin. Processes Geophys. 19, 449, 2012
Oceanicflows
LAGRANGIAN DESCRIPTORS
EXAMPLE: INVARIANT MANIFOLDS IN DUFFING EQ.
J. A. Jiménez Madrid, A. M. Mancho, Chaos 19, 013111 2009
LAGRANGIAN DESCRIPTORS: P-NORM
C. Lopesino, F. Balibrea-Iniesta, V. J. García-Garrido, S. Wigginsand A. M. Mancho, Int. J. Bifur. and Chaos 27, 1730001 (2017)
INVARIANT MANIFOLDS = SINGULARITIES
C. Lopesino, F. Balibrea-Iniesta, V. J. García-Garrido, S. Wigginsand A. M. Mancho, Int. J. Bifur. and Chaos 27, 1730001 (2017)
INVARIANT MANIFOLDS = SINGULARITIES
C. Lopesino, F. Balibrea-Iniesta, V. J. García-Garrido, S. Wigginsand A. M. Mancho, Int. J. Bifur. and Chaos 27, 1730001 (2017)
Results
Results1. System with 2 dof2. Saddle-node bifurcation3. System with 3 dof
𝐸𝐸 = 1500𝑐𝑐𝑐𝑐−1POINCARÉ SUPERFICIE OF SECTION
𝐸𝐸 = 1500𝑐𝑐𝑐𝑐−1 𝐸𝐸 = 2500𝑐𝑐𝑐𝑐−1POINCARÉ SUPERFICIE OF SECTION
SSP𝐸𝐸 = 3500𝑐𝑐𝑐𝑐−1
SSP Lagrangian Descr.𝐸𝐸 = 3500𝑐𝑐𝑐𝑐−1
SSP Lagrangian Descr.𝐸𝐸 = 3500𝑐𝑐𝑐𝑐−1
SSP Lagrangian Descr.𝐸𝐸 = 4000𝑐𝑐𝑐𝑐−1
SSP Lagrangian Descr.𝐸𝐸 = 4000𝑐𝑐𝑐𝑐−1
SSP Lagrangian Descr.𝐸𝐸 = 4000𝑐𝑐𝑐𝑐−1
SSP Lagrangian Descr.𝐸𝐸 = 4500𝑐𝑐𝑐𝑐−1
SSP Lagrangian Descr.𝐸𝐸 = 4500𝑐𝑐𝑐𝑐−1
SSP Lagrangian Descr.𝐸𝐸 = 3500𝑐𝑐𝑐𝑐−1
SSP Lagrangian Descr.𝐸𝐸 = 3500𝑐𝑐𝑐𝑐−1
POTENTIAL ENERGY SURFACENC
Li Rθ
Periodic orbitsCenter: marginallly stableLeft: StableRight: Unstable
POTENTIAL ENERGY SURFACENC
Li Rθ
Periodic orbitsCenter: marginallly stableLeft: StableRight: Unstable
POTENTIAL ENERGY SURFACENC
Li Rθ
Periodic orbitsCenter: marginallly stableLeft: StableRight: Unstable
POTENTIAL ENERGY SURFACENC
Li Rθ
Periodic orbitsCenter: marginallly stableLeft: StableRight: Unstable
POTENTIAL ENERGY SURFACENC
Li Rθ
Periodic orbitsCenter: marginallly stableLeft: StableRight: Unstable
SADDLE-NODEBIFURCATION
𝐸𝐸 = 3000 𝑐𝑐𝑐𝑐−1
𝑝𝑝 = 0.4τ = 2 . 104
SADDLE-NODEBIFURCATION
𝐸𝐸 = 3200 𝑐𝑐𝑐𝑐−1
𝑝𝑝 = 0.4τ = 2 . 104
SADDLE-NODEBIFURCATION
𝐸𝐸 ≃ 3441 𝑐𝑐𝑐𝑐−1
𝑝𝑝 = 0.4τ = 2 . 104
SADDLE-NODEBIFURCATION
𝐸𝐸 = 3700 𝑐𝑐𝑐𝑐−1
𝑝𝑝 = 0.4τ = 2 . 104
SADDLE-NODEBIFURCATION
𝐸𝐸 ≃ 4162 𝑐𝑐𝑐𝑐−1
𝑝𝑝 = 0.4τ = 2 . 104
SADDLE-NODEBIFURCATION
𝐸𝐸 ≃ 4162 𝑐𝑐𝑐𝑐−1
𝑝𝑝 = 0.4τ = 2 . 104
SADDLE-NODEBIFURCATION
𝐸𝐸 ≃ 4162 𝑐𝑐𝑐𝑐−1
𝑝𝑝 = 0.4τ = 2 . 104
N3 dof
RθLi
rC
N3 dof
RθLi
rC
𝑝𝑝 = 0.4τ = 2 . 104
3-DOF SYSTEM
3-DOF SYSTEM
The more initial kinetic energy is set in the r-dof, themore regular the phase space is in the other dofs’s
𝐸𝐸 = 3500 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 0 𝑐𝑐𝑐𝑐−1
𝐸𝐸 = 3500 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 1500 𝑐𝑐𝑐𝑐−1
CONCLUSIONS
CONCLUSIONSLagrangiandescriptors allow theidentification of theinvariant manifolds in molecular systems
SPONSORS
Fabio Revuelta
Dynamics Days Central Asia VI 5 June 2020
NEW PERSPECTIVES INTO THE CHAOTICDYNAMICS IN MOLECULES
Grupo de Sistemas ComplejosUniversidad Politécnica de Madrid (Spain)
POINCARÉ SURFACE OF SECTION
INFLUENCE OF THE INTEGRATION TIME
τ must be longenough… but not too
much
LAGRANGIANDESCRIPTORS𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆τ = 103
LAGRANGIANDESCRIPTORS𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆τ = 104
LAGRANGIANDESCRIPTORS𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆τ = 2 . 104
LAGRANGIANDESCRIPTORS𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆τ = 105
INFLUENCE OF THE VALUEOF P IN THE NORM
Image source:
LAGRANGIANDESCRIPTORS𝑝𝑝 = 0.1
τ = 2 . 104
LAGRANGIANDESCRIPTORS𝑝𝑝 = 0.4
τ = 2 . 104
LAGRANGIANDESCRIPTORS𝑝𝑝 = 0.6
τ = 2 . 104
LAGRANGIANDESCRIPTORS
𝑝𝑝 = 1τ = 2 . 104
3 DOF
𝐸𝐸 = 3500 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 1500 𝑐𝑐𝑐𝑐−1
𝐸𝐸 = 4000 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 1500 𝑐𝑐𝑐𝑐−1
𝐸𝐸 = 4500 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 1500 𝑐𝑐𝑐𝑐−1
𝐸𝐸 = 3500 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 1000 𝑐𝑐𝑐𝑐−1
𝐸𝐸 = 4000 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 1000 𝑐𝑐𝑐𝑐−1
𝐸𝐸 = 4500 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 1000 𝑐𝑐𝑐𝑐−1
𝐸𝐸 = 3500 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 0 𝑐𝑐𝑐𝑐−1
𝐸𝐸 = 3500 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 0 𝑐𝑐𝑐𝑐−1
𝐸𝐸 = 4000 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 0 𝑐𝑐𝑐𝑐−1
𝐸𝐸 = 4500 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 0 𝑐𝑐𝑐𝑐−1