Henon Heiles Research Articles

Physics guided neural networks for modelling of non-linear dynamics

The success of the current wave of artificial intelligence can be partly attributed to deep neural networks, which have proven to be very effective in learning complex patterns from large datasets with minimal human intervention. However, it is difficult to train these models on complex dynamical systems from data alone due to their low data efficiency and sensitivity to hyperparameters and initialisation. This work demonstrates that injection of partially known information at an intermediate layer in a DNN can improve model accuracy, reduce model uncertainty, and yield improved convergence during the training. The value of these physics-guided neural networks has been demonstrated by learning the dynamics of a wide variety of nonlinear dynamical systems represented by five well-known equations in nonlinear systems theory: the Lotka–Volterra, Duffing, Van der Pol, Lorenz, and Henon–Heiles systems.

Distribution of stable islands within chaotic areas in the non-hyperbolic and hyperbolic regimes in the Hénon–Heiles system

We provide rigorous computer-assisted proofs of the existence of different dynamical objects, like stable families of periodic orbits, bifurcations and stable invariant tori around them, in the paradigmatic Henon–Heiles system. There are in the literature a large number of articles with numerical simulations on this system, and other open Hamiltonians, but only a few give a rigorous guarantee of simulations. In this article, we present the necessary link between the numerical simulations and the mathematical structure of the system, since it is relevant to provide evidence of the existence of some of the different objects detected numerically to evaluate the quality of the numerical results. Remarkably, we present a proof of the existence of stable regions in the non-hyperbolic and hyperbolic regimes classically established for the Henon–Heiles system. In particular, we prove the important results of the existence of bounded stable regular regions located within the escape region, far from the regime of the KAM islands, which are called “safe regions”.

Liouvillian integrability of the three-dimensional generalized Hénon–Heiles Hamiltonian

In this paper, we report about three cases of integrability in sense of Liouville for three-dimensional generalized Henon–Heiles Hamiltonian. This also allow to get explicitly integrals of motions for each case. On the other hand, this paper investigates the phase space structure numerically with Poincare surfaces of section and 3D projections which allow to verify that the analytical results are in agreement with the computations.

Guaranteed a posteriori error bounds for low-rank tensor approximate solutions

Abstract We propose a guaranteed and fully computable upper bound on the energy norm of the error in low-rank tensor train (TT) approximate solutions of (possibly) high-dimensional reaction–diffusion problems. The error bound is obtained from Euler–Lagrange equations for a complementary flux reconstruction problem, which are solved in the low-rank TT representation using the block alternating linear scheme. This bound is guaranteed to be above the energy norm of the total error, including the discretization error, the tensor approximation error and the error in the solver of linear algebraic equations, although quadrature errors, in general, can pollute its evaluation. Numerical examples with the Poisson equation and the Schrödinger equation with the Henon–Heiles potential in up to 40 dimensions are presented to illustrate the efficiency of this approach.

Hamiltonian thermodynamics in the extended phase space: a unifying theory for non-linear molecular dynamics and classical thermodynamics

The Hamiltonian theory of thermodynamics for reversible and irreversible processes is presented in detail. Particularly, it is demonstrated that non-linear molecular dynamics and thermodynamics of equilibrium and non-equilibrium processes can merge in a single dynamical theory by constructing a homogeneous of first degree in momenta Hamiltonian function on the extended thermodynamic state space and in the entropy representation. The geometrical properties of the physical equilibrium state manifold are discussed and results of numerical experiments with the Henon–Heiles model that dissipates in a simple thermodynamic system with multiple friction parameters are presented. It is well known that this classical paradigm carries most of the non-linear dynamical characteristics of generic molecular systems. The construction of homogeneous Hamiltonians for composite thermodynamic systems is also outlined in the appendixes, where the theory of Hamiltonian thermodynamics is developed in current differential geometry terms.

Reeb’s Theorem and Periodic Orbits for a Rotating Hénon–Heiles Potential

We apply Reeb’s theorem to prove the existence of periodic orbits in the rotating Henon–Heiles system. To this end, a sort of detuned normal form is calculated that yields a reduced system with at most four non degenerate equilibrium points. Linear stability and bifurcations of equilibrium solutions mimic those for periodic solutions of the original system. We also determine heteroclinic connections that can account for transport phenomena.

Hidden Nambu mechanics II: Quantum/semiclassical dynamics

Abstract Nambu mechanics is a generalized Hamiltonian dynamics characterized by an extended phase space and multiple Hamiltonians. In a previous paper [Prog. Theor. Exp. Phys. 2013, 073A01 (2013)] we revealed that the Nambu mechanical structure is hidden in Hamiltonian dynamics, that is, the classical time evolution of variables including redundant degrees of freedom can be formulated as Nambu mechanics. In the present paper we show that the Nambu mechanical structure is also hidden in some quantum or semiclassical dynamics, that is, in some cases the quantum or semiclassical time evolution of expectation values of quantum mechanical operators, including composite operators, can be formulated as Nambu mechanics. We present a procedure to find hidden Nambu structures in quantum/semiclassical systems of one degree of freedom, and give two examples: the exact quantum dynamics of a harmonic oscillator, and semiclassical wave packet dynamics. Our formalism can be extended to many-degrees-of-freedom systems; however, there is a serious difficulty in this case due to interactions between degrees of freedom. To illustrate our formalism we present two sets of numerical results on semiclassical dynamics: from a one-dimensional metastable potential model and a simplified Henon–Heiles model of two interacting oscillators.

Entropy and Mixing Entropy for Weakly Nonlinear Mechanical Vibrating Systems.

In this paper, we examine Khinchin’s entropy for two weakly nonlinear systems of oscillators. We study a system of coupled Duffing oscillators and a set of Henon–Heiles oscillators. It is shown that the general method of deriving the Khinchin’s entropy for linear systems can be modified to account for weak nonlinearities. Nonlinearities are modeled as nonlinear springs. To calculate the Khinchin’s entropy, one needs to obtain an analytical expression of the system’s phase volume. We use a perturbation method to do so, and verify the results against the numerical calculation of the phase volume. It is shown that such an approach is valid for weakly nonlinear systems. In an extension of the author’s previous work for linear systems, a mixing entropy is defined for these two oscillators. The mixing entropy is the result of the generation of entropy when two systems are combined to create a complex system. It is illustrated that mixing entropy is always non-negative. The mixing entropy provides insight into the energy behavior of each system. The limitation of statistical energy analysis motivates this study. Using the thermodynamic relationship of temperature and entropy, and Khinchin’s entropy, one can define a vibrational temperature. Vibrational temperature can be used to derive the power flow proportionality, which is the backbone of the statistical energy analysis. Although this paper is motivated by statistical energy analysis application, it is not devoted to the statistical energy analysis of nonlinear systems.

On the families of fractional dynamical models

In this paper, we reveal the uncertainty and its fractional generalized Hamiltonian representation for the fractional and nonlinear problem and find general methods of constructing a family of fractional dynamical models. By using the definition of combined fractional derivative, we construct a unified fractional generalized Hamiltonian equation, a fractional generalized Hamiltonian equation with combined Riemann–Liouville derivative, and a fractional generalized Hamiltonian equation with combined Caputo derivative; also, as special cases, under the different definitions of fractional derivatives, we, respectively, obtain a series of different kinds of fractional generalized Hamiltonian equations. In particular, we present a new concept of the family of fractional dynamical models, and it is found that, using the fractional generalized Hamiltonian method, we can construct a series of families of fractional dynamical models. And then, as the new method’s application, we construct three new kinds of families of fractional dynamical models, which include a family of fractional Lorentz–Dirac models, a family of fractional Lotka–Volterra models and a family of fractional Henon–Heiles models.

The Henon–Heiles system defined on Lie-algebraically deformed Galilei spacetime

In this paper, we provide the Henon–Heiles system defined on Lie-algebraically deformed non-relativistic spacetime with the commutator of two spatial directions proportional to time. Particularly, we demonstrate that in such a model the total energy is not conserved and for this reason the role of control parameter is taken by the initial energy value [Formula: see text]. Besides, we show that in contrast with the commutative case, for chosen values of deformation parameter [Formula: see text], there appears chaos in the system for initial total energies [Formula: see text] below the threshold [Formula: see text].

Probabilistic evolution theory for the solution of explicit autonomous ordinary differential equations: squarified telescope matrices

Probabilistic evolution theory (PREVTH) is used for the solution of initial value problems of first order explicit autonomous ordinary differential equation sets with second degree multinomial right hand side functions. It is an approximation method based on Kronecker power series: a rewriting of multivariate Taylor series using matrices having certain flexible parameters. Kronecker power series have matrices which are called telescope matrices: \(n \times n^{j+1}\) matrices where j is the index of summation. The additive terms of each telescope matrix is formed through Kronecker product from both sides by Kronecker powers of identity matrices. Recently, squarification is proposed in order to avoid the growing of the matrices in size at each additive term of the series. This paper explains the squarification procedure: the procedure used in order to avoid Kronecker multiplications within PREVTH so that the sizes of the matrices do not grow and so that the amount of necessary computation is reduced. The recursion between squarified matrices is also given. As a numerical application, the solution of a Henon–Heiles system is provided.

NuSol — Numerical solver for the 3D stationary nuclear Schrödinger equation

The classification of short hydrogen bonds depends on several factors including the shape and energy spacing between the nuclear eigenstates of the hydrogen.Here, we describe the NuSol program in which three classes of algorithms were implemented to solve the 1D, 2D and 3D time independent nuclear Schrödinger equation. The Schrödinger equation was solved using the finite differences based Numerov’s method which was extended to higher dimensions, the more accurate pseudo-spectral Chebyshev collocation method and the sinc discrete variable representation by Colbert and Miller. NuSol can be applied to solve the Schrödinger equation for arbitrary analytical or numerical potentials with focus on nuclei bound by the potential of their molecular environment. We validated the methods against literature values for the 2D Henon–Heiles potential, the 3D linearly coupled sextic oscillators and applied them to study hydrogen bonding in the malonaldehyde derivate 4-cyano-2,2,6,6-tetramethyl-3,5-heptanedione.With NuSol, the extent of nuclear delocalization in a given molecular potential can directly be calculated without relying on linear reaction coordinates in 3D molecular space. Program summaryProgram title: NuSolCatalogue identifier: AEXO_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEXO_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GPL 3.0No. of lines in distributed program, including test data, etc.: 195332No. of bytes in distributed program, including test data, etc.: 4489808Distribution format: tar.gzProgramming language: Python 2.7, C.Computer: PC.Operating system: Linux.RAM: 0.1–40 GBClassification: 16.1.External routines: FEAST v2.1 [1][2] (included in the distribution file).Nature of problem: Solving the 3D nuclear Schrödinger EquationSolution method: Grid solver based on Numerov’s method, Chebyshev collocation and sinc() DVR for 1D/2D/3D potential gridsRunning time: System dependent

An improved Lobatto discrete variable representation by a phase optimisation and variable mapping method

The Lobatto discrete variable representation (LDVR) proposed by Manoloupolos and Wyatt (1988) has unique features but has not been generally applied in the field of chemical dynamics. Instead, it has popular application in solving atomic physics problems, in combining with the finite element method (FE-DVR), due to its inherent abilities for treating the Coulomb singularity in spherical coordinates. In this work, an efficient phase optimisation and variable mapping procedure is proposed to improve the grid efficiency of the LDVR/FE-DVR method, which makes it not only be competing with the popular DVR methods, such as the Sinc-DVR, but also keep its advantages for treating with the Coulomb singularity. The method is illustrated by calculations for one-dimensional Coulomb potential, and the vibrational states of one-dimensional Morse potential, two-dimensional Morse potential and two-dimensional Henon–Heiles potential, which prove the efficiency of the proposed scheme and promise more general applications of the LDVR/FE-DVR method.

Stability for manifolds of the equilibrium state of fractional Birkhoffian systems

In the paper, we present a new stability theory of fractional dynamics, i.e., the stability for manifolds of equilibrium state of a fractional Birkhoffian system, in terms of Riesz derivatives, and explore its applications. For an autonomous fractional Birkhoffian system, the gradient representation and second-order gradient representation are studied, and the conditions under which the system can be considered as a gradient system and a second-order gradient system are given, respectively. Then, equilibrium equations, perturbation equations and first approximate equations are obtained, and the stability theorem for manifolds of equilibrium state of the general autonomous system is used to a fractional Birkhoffian system, and three criterions on the stability for manifolds of the equilibrium state of the system are investigated. As special cases of the fractional Birkhoffian method, the three criterions can also be applied to a traditional Birkhoffian system, a fractional Hamiltonian system and a classical Hamiltonian system, respectively. As applications, by using the fractional Birkhoffian method of this paper, we construct three kinds of fractional dynamical models, which include a four-dimensional fractional dynamical model, a fractional Henon–Heiles model and a fractional Lorentz–Dirac model, and we explore the stability for manifolds of the equilibrium state of these models, respectively. This work provides a general method for studying the dynamic stability of an actual fractional dynamical model that is related to science and engineering.

Fractional Birkhoffian mechanics

In this paper, we present a new fractional dynamical theory, i.e., the dynamics of a Birkhoffian system with fractional derivatives (the fractional Birkhoffian mechanics), which gives a general method for constructing a fractional dynamical model of the actual problem. By using the definition of combined fractional derivative, we present a unified fractional Pfaff action and a unified fractional Pfaff–Birkhoff principle, and give four kinds of fractional Pfaff–Birkhoff principles under the different definitions of fractional derivative. And then, by using the fractional Pfaff–Birkhoff principles, we establish a series of fractional Birkhoffian equations with different fractional derivatives and construct the tensor representation of autonomous fractional Birkhoffian equations. Further, we study the relationship among the fractional Bikhoffian system, the fractional Hamiltonian system and the fractional Lagrangian system and give the transformation conditions. And furthermore, as applications of the fractional Birkhoffian method, we construct five kinds of fractional dynamical models, which include the fractional Lotka biochemical oscillator model, the fractional Whittaker model, the fractional Hojman–Urrutia model, the fractional Henon–Heiles model and the fractional Emden model.

Lie algebraic structure and generalized Poisson conservation law for fractional generalized Hamiltonian systems

Based on definition of Riemann–Liouville fractional derivatives, we find that the fractional generalized Hamiltonian system possesses consistent algebraic structure and Lie algebraic structure, and the generalized Poisson conservation law of the fractional generalized Hamiltonian system is investigated. As special cases of this paper, the conditions under which a fractional generalized Hamiltonian system can be reduced to a generalized Hamiltonian system and a classical Hamiltonian system are given, and the Lie algebraic structure and Poisson conservation law of these special dynamical systems are obtained, respectively. Then, by using the method and results of this paper, we construct three kinds of new fractional dynamical model, i.e. a fractional Henon–Heiles model, a fractional Euler–Poinsot model of a rigid body and a fractional Volterra model of the three species groups, and their Poisson conserved quantities are obtained.

Stability for manifolds of equilibrium states of fractional generalized Hamiltonian systems

For a fractional generalized Hamiltonian system, in terms of Riesz derivatives, stability theory for the manifolds of equilibrium states is presented. The gradient representation and second order gradient representation of a fractional generalized Hamiltonian system are studied, and the conditions under which the system can be considered as a gradient system and a second order gradient system are given, respectively. Then, equilibrium equations, disturbance equations, and first approximate equations of a fractional generalized Hamiltonian system are obtained. A theorem for the stability of the manifolds of equilibrium states of the general autonomous system is used to a fractional generalized Hamiltonian system, and three propositions on the stability of the manifolds of equilibrium states of the system are investigated. As the special cases of this article, the conditions which a fractional generalized Hamiltonian system can be reduced to a generalized Hamiltonian system, a fractional Hamiltonian system and a Hamiltonian system are given, respectively, and the stability theory for the manifolds of equilibrium states of these systems are obtained. Further, a fractional dynamical system and a fractional Volterra model of the three species groups are given to illustrate the method and results of the application. Finally, by using the method in this paper, we construct a new kind of fractional dynamical model, i.e. the fractional Henon–Heiles model, and we study its stability of the manifolds of equilibrium states.

Regular and chaotic motion in Henon Heiles like Hamiltonian

The dynamics of a system subjected to a potential equal to the sum of the Hénon--Heiles potential and that of hydrogen in an electric field was studied. The 4 Hamilton's equations of motion follow from the Hamiltonian and they were integrated numerically using the Runge-Kutta fourth order method. The Poincaré surface of a section fixed at x = 0 and px> 0 was used to reduce the phase space to a 2-dimensional plane. The analysis of the Poincaré surface, the Lyapunov exponent, and the autocorrelation shows that as the constant of motion, E, increases from 0.30 to 0.45, the dynamics makes a transition from periodic and quasi-periodic to chaotic motions.

Mutual influence of locality and chaotic dynamics on quantum controllability

Quantum control tasks are classified either as classical-like or as quantum requiring interference of pathways. We study the generation of interference pathways and relate them to the fidelity of the control target at a fixed time for various tasks. The model drift Hamiltonian studied is the two-dimensional Henon-Heiles (HH) potential. This system shows regular classical dynamics for low energies and chaotic dynamics for higher energies. A control operator supported by the whole momentum space and therefore connecting the entire Hilbert phase space is a random spiky potential. The other extreme is a smooth control potential. Intermediate cases are obtained by filtering the random spiky potential in momentum space. The fidelity of achieving a control task was related to the connectivity in phase space of the control operators. Typical quantum tasks such as generating a superposition of generalized coherent states rely on interfering pathways. For these cases the nonlinearity in the drift or control Hamiltonian is a necessary requirement for creating interferences. Control over rapidly diverging components of the wave function is achieved by the use of highly nonlocal control operators. Quantum control under chaotic drift was found to give a better yield than control under regular dynamics for such cases. For classical tasks we study the transformation of an initial generalized coherent state to another one. The best fidelity is obtained for regular or harmonic regions of the potential and smooth control operators. The approach to the classical limit is checked by decreasing the effective value of $\ensuremath{\hbar}$. Control under both quantum and classical tasks suffered from the decrease of $\ensuremath{\hbar}$ and the approach to classical proximity. Classical control tasks which rely heavily on maintaining a generalized coherent state throughout the evolution were found to be dysfunctional and lead to a completely uncontrolled situation once the classical chaos starts to appear.

Separation of variables for the generalized Henon–Heiles system and system with quartic potential

We consider two well-known integrable systems on the plane using the concept of natural Poisson bivectors on Riemannian manifolds. A geometric approach to construction of variables of separation and separated relations for the generalized Henon–Heiles system and the generalized system with quartic potential is discussed in detail.