new perspectives into the chaotic dynamics in molecules · 2020. 5. 22. · fabio revuelta....

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