NEW PERSPECTIVES INTO THE CHAOTIC DYNAMICS IN MOLECULES · 2020. 5. 22. · Fabio Revuelta....
Transcript of NEW PERSPECTIVES INTO THE CHAOTIC DYNAMICS IN MOLECULES · 2020. 5. 22. · Fabio Revuelta....
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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)
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NOLINEAL 20-21
12th International Conference onNonlinear Mathematics and Physics
30 June – 2 July 2021
Madrid (Spain)
https://www.gsc.upm.es/nolineal2020
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NOLINEAL 20-21
12th International Conference onNonlinear Mathematics and Physics
30 June – 2 July 2021
Madrid (Spain)
https://www.gsc.upm.es/nolineal2020
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OUTLINE
IntroductionSystemMethodologyResultsConclusions
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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
…
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ESSENTIALS
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ESSENTIALS
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ESSENTIALSReference trajectory
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ESSENTIALSReference trajectory
Nearbytrajectory 1
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ESSENTIALSReference trajectory
Nearbytrajectory 1
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ESSENTIALSReference trajectory
Nearbytrajectory 1
Nearbytrajectory 2
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ESSENTIALSReference trajectory
Nearbytrajectory 1
Nearbytrajectory 2
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ESSENTIALSReference trajectory
Nearbytrajectory 1
Nearbytrajectory 2
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ESSENTIALSReference trajectory
Nearbytrajectory 1
Nearbytrajectory 2
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ESSENTIALSReference trajectory
Nearbytrajectory 1
Nearbytrajectory 2
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ESSENTIALSReference trajectory
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ESSENTIALSReference trajectory
Lagrangian descriptors: allthe information encoded in the trajectory itself!
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SYSTEM
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WHY MOLECULES???
1. Atomic and molecular systems: excellentplatforms for application of dynamicalsystems theory
2. ⇑⇑⇑ anharmonic3. “Simple” (2 dof)
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SYSTEM
N
Li
C
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SYSTEM
N
Li
C
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SYSTEM
N
Li
C
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SYSTEM
N
Li
C
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SYSTEM
N
Li
C
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SYSTEM
N
Li
C
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SYSTEM
1. 3 atoms: 9 dof2. No rotation: - 6 dof3. Coordinates: R, θ, r NC
Li
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SYSTEM
N
Li
C
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SYSTEM
N
RLi
C
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SYSTEM
N
RθLi
C
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SYSTEM
N
RθLi
Cr
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SYSTEM
N
RθLi
Cr≃req
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SYSTEM
N
RθLi
Cr≃req
(2 DOF)
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HAMILTONIAN
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HAMILTONIAN
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HAMILTONIAN
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HAMILTONIAN
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HAMILTONIAN
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HAMILTONIAN
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HAMILTONIAN
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HAMILTONIAN
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POTENTIAL ENERGY SURFACENC
Li Rθ
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POTENTIAL ENERGY SURFACENC
Li Rθ
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POTENTIAL ENERGY SURFACENC
Li Rθ
Absoluteminimum
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POTENTIAL ENERGY SURFACENC
Li Rθ
LiC N
Absoluteminimum
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POTENTIAL ENERGY SURFACENC
Li Rθ
LiC N
Relativeminimum Absolute
minimum
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POTENTIAL ENERGY SURFACENC
Li Rθ
NLi C LiC N
Relativeminimum Absolute
minimum
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POTENTIAL ENERGY SURFACENC
Li Rθ
NLi C LiC N
Relativeminimum Absolute
minimumMEP
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POTENTIAL ENERGY SURFACENC
Li Rθ
NLi C LiC N
Relativeminimum Absolute
minimumSaddle
MEP
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POTENTIAL ENERGY SURFACENC
Li
r
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Methodology
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LAGRANGIAN DESCRIPTORS
Figure: C. Mendoza, A. M. Mancho,Nonlin. Processes Geophys. 19, 449, 2012
Oceanicflows
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LAGRANGIAN DESCRIPTORS
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EXAMPLE: INVARIANT MANIFOLDS IN DUFFING EQ.
J. A. Jiménez Madrid, A. M. Mancho, Chaos 19, 013111 2009
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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)
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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)
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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)
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Results
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Results1. System with 2 dof2. Saddle-node bifurcation3. System with 3 dof
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𝐸𝐸 = 1500𝑐𝑐𝑐𝑐−1POINCARÉ SUPERFICIE OF SECTION
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𝐸𝐸 = 1500𝑐𝑐𝑐𝑐−1 𝐸𝐸 = 2500𝑐𝑐𝑐𝑐−1POINCARÉ SUPERFICIE OF SECTION
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SSP𝐸𝐸 = 3500𝑐𝑐𝑐𝑐−1
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SSP Lagrangian Descr.𝐸𝐸 = 3500𝑐𝑐𝑐𝑐−1
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SSP Lagrangian Descr.𝐸𝐸 = 3500𝑐𝑐𝑐𝑐−1
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SSP Lagrangian Descr.𝐸𝐸 = 4000𝑐𝑐𝑐𝑐−1
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SSP Lagrangian Descr.𝐸𝐸 = 4000𝑐𝑐𝑐𝑐−1
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SSP Lagrangian Descr.𝐸𝐸 = 4000𝑐𝑐𝑐𝑐−1
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SSP Lagrangian Descr.𝐸𝐸 = 4500𝑐𝑐𝑐𝑐−1
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SSP Lagrangian Descr.𝐸𝐸 = 4500𝑐𝑐𝑐𝑐−1
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SSP Lagrangian Descr.𝐸𝐸 = 3500𝑐𝑐𝑐𝑐−1
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SSP Lagrangian Descr.𝐸𝐸 = 3500𝑐𝑐𝑐𝑐−1
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POTENTIAL ENERGY SURFACENC
Li Rθ
Periodic orbitsCenter: marginallly stableLeft: StableRight: Unstable
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POTENTIAL ENERGY SURFACENC
Li Rθ
Periodic orbitsCenter: marginallly stableLeft: StableRight: Unstable
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POTENTIAL ENERGY SURFACENC
Li Rθ
Periodic orbitsCenter: marginallly stableLeft: StableRight: Unstable
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POTENTIAL ENERGY SURFACENC
Li Rθ
Periodic orbitsCenter: marginallly stableLeft: StableRight: Unstable
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POTENTIAL ENERGY SURFACENC
Li Rθ
Periodic orbitsCenter: marginallly stableLeft: StableRight: Unstable
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SADDLE-NODEBIFURCATION
𝐸𝐸 = 3000 𝑐𝑐𝑐𝑐−1
𝑝𝑝 = 0.4τ = 2 . 104
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SADDLE-NODEBIFURCATION
𝐸𝐸 = 3200 𝑐𝑐𝑐𝑐−1
𝑝𝑝 = 0.4τ = 2 . 104
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SADDLE-NODEBIFURCATION
𝐸𝐸 ≃ 3441 𝑐𝑐𝑐𝑐−1
𝑝𝑝 = 0.4τ = 2 . 104
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SADDLE-NODEBIFURCATION
𝐸𝐸 = 3700 𝑐𝑐𝑐𝑐−1
𝑝𝑝 = 0.4τ = 2 . 104
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SADDLE-NODEBIFURCATION
𝐸𝐸 ≃ 4162 𝑐𝑐𝑐𝑐−1
𝑝𝑝 = 0.4τ = 2 . 104
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SADDLE-NODEBIFURCATION
𝐸𝐸 ≃ 4162 𝑐𝑐𝑐𝑐−1
𝑝𝑝 = 0.4τ = 2 . 104
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SADDLE-NODEBIFURCATION
𝐸𝐸 ≃ 4162 𝑐𝑐𝑐𝑐−1
𝑝𝑝 = 0.4τ = 2 . 104
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N3 dof
RθLi
rC
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N3 dof
RθLi
rC
𝑝𝑝 = 0.4τ = 2 . 104
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3-DOF SYSTEM
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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
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𝐸𝐸 = 3500 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 0 𝑐𝑐𝑐𝑐−1
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𝐸𝐸 = 3500 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 1500 𝑐𝑐𝑐𝑐−1
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CONCLUSIONS
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CONCLUSIONSLagrangiandescriptors allow theidentification of theinvariant manifolds in molecular systems
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SPONSORS
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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)
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POINCARÉ SURFACE OF SECTION
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INFLUENCE OF THE INTEGRATION TIME
τ must be longenough… but not too
much
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LAGRANGIANDESCRIPTORS𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆τ = 103
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LAGRANGIANDESCRIPTORS𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆τ = 104
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LAGRANGIANDESCRIPTORS𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆τ = 2 . 104
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LAGRANGIANDESCRIPTORS𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆τ = 105
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INFLUENCE OF THE VALUEOF P IN THE NORM
Image source:
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LAGRANGIANDESCRIPTORS𝑝𝑝 = 0.1
τ = 2 . 104
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LAGRANGIANDESCRIPTORS𝑝𝑝 = 0.4
τ = 2 . 104
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LAGRANGIANDESCRIPTORS𝑝𝑝 = 0.6
τ = 2 . 104
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LAGRANGIANDESCRIPTORS
𝑝𝑝 = 1τ = 2 . 104
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3 DOF
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𝐸𝐸 = 3500 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 1500 𝑐𝑐𝑐𝑐−1
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𝐸𝐸 = 4000 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 1500 𝑐𝑐𝑐𝑐−1
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𝐸𝐸 = 4500 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 1500 𝑐𝑐𝑐𝑐−1
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𝐸𝐸 = 3500 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 1000 𝑐𝑐𝑐𝑐−1
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𝐸𝐸 = 4000 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 1000 𝑐𝑐𝑐𝑐−1
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𝐸𝐸 = 4500 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 1000 𝑐𝑐𝑐𝑐−1
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𝐸𝐸 = 3500 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 0 𝑐𝑐𝑐𝑐−1
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𝐸𝐸 = 3500 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 0 𝑐𝑐𝑐𝑐−1
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𝐸𝐸 = 4000 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 0 𝑐𝑐𝑐𝑐−1
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𝐸𝐸 = 4500 𝑐𝑐𝑐𝑐−1
𝑇𝑇𝐶𝐶𝐶𝐶𝑘𝑘𝑘𝑘𝑘𝑘 = 0 𝑐𝑐𝑐𝑐−1