scholarly journals Orbit Classification and Sensitivity Analysis in Dynamical Systems Using Surrogate Models

2021 ◽  
Vol 2 (1) ◽  
pp. 5
Author(s):  
Katharina Rath ◽  
Christopher G. Albert ◽  
Bernd Bischl ◽  
Udo von Toussaint
Keyword(s):  
Sensitivity Analysis ◽  
Phase Space ◽  
Dynamical Systems ◽  
Hamiltonian Systems ◽  
Initial Conditions ◽  
Plasma Boundary ◽  
Surrogate Models ◽  
Alpha Particles ◽  
Hamiltonian Flow ◽  
Chaotic Orbits

Dynamics of many classical physics systems are described in terms of Hamilton’s equations. Commonly, initial conditions are only imperfectly known. The associated volume in phase space is preserved over time due to the symplecticity of the Hamiltonian flow. Here we study the propagation of uncertain initial conditions through dynamical systems using symplectic surrogate models of Hamiltonian flow maps. This allows fast sensitivity analysis with respect to the distribution of initial conditions and an estimation of local Lyapunov exponents (LLE) that give insight into local predictability of a dynamical system. In Hamiltonian systems, LLEs permit a distinction between regular and chaotic orbits. Combined with Bayesian methods we provide a statistical analysis of local stability and sensitivity in phase space for Hamiltonian systems. The intended application is the early classification of regular and chaotic orbits of fusion alpha particles in stellarator reactors. The degree of stochastization during a given time period is used as an estimate for the probability that orbits of a specific region in phase space are lost at the plasma boundary. Thus, the approach offers a promising way to accelerate the computation of fusion alpha particle losses.

scholarly journals Decay of Hamiltonian Breathers Under Dissipation

2020 ◽  
Vol 380 (1) ◽  
pp. 71-102
Author(s):  
Jean-Pierre Eckmann ◽  
C. Eugene Wayne
Keyword(s):  
Phase Space ◽  
Systems Theory ◽  
Hamiltonian Systems ◽  
Initial Conditions ◽  
Asymptotic Estimates ◽  
Discrete Nonlinear Schrödinger Equation ◽  
Rate Of Evolution ◽  
Energy Flows ◽  
The Family ◽  
Almost All

Abstract We study metastable behavior in a discrete nonlinear Schrödinger equation from the viewpoint of Hamiltonian systems theory. When there are $$n<\infty $$ n < ∞ sites in this equation, we consider initial conditions in which almost all the energy is concentrated in one end of the system. We are interested in understanding how energy flows through the system, so we add a dissipation of size $$\gamma $$ γ at the opposite end of the chain, and we show that the energy decreases extremely slowly. Furthermore, the motion is localized in the phase space near a family of breather solutions for the undamped system. We give rigorous, asymptotic estimates for the rate of evolution along the family of breathers and the width of the neighborhood within which the trajectory is confined.

scholarly journals Hamiltonian flow over saddles for exploring molecular phase space structures

2018 ◽  
Vol 376 (2115) ◽  
pp. 20170148 ◽  
Author(s):  
Stavros C. Farantos
Keyword(s):  
Phase Space ◽  
Potential Energy Surfaces ◽  
Initial Conditions ◽  
Space Structures ◽  
Hamiltonian Flow ◽  
Molecular Configuration ◽  
Geometrical Structures ◽  
Classical Dynamical System ◽  
Energy Surfaces ◽  
Alanine Dipeptide

Despite using potential energy surfaces, multivariable functions on molecular configuration space, to comprehend chemical dynamics for decades, the real happenings in molecules occur in phase space, in which the states of a classical dynamical system are completely determined by the coordinates and their conjugate momenta. Theoretical and numerical results are presented, employing alanine dipeptide as a model system, to support the view that geometrical structures in phase space dictate the dynamics of molecules, the fingerprints of which are traced by following the Hamiltonian flow above saddles. By properly selecting initial conditions in alanine dipeptide, we have found internally free rotor trajectories the existence of which can only be justified in a phase space perspective. This article is part of the theme issue ‘Modern theoretical chemistry’.

scholarly journals THE STRUCTURE OF PHASE SPACE CLOSE TO FIXED POINTS IN A 4D SYMPLECTIC MAP

2013 ◽  
Vol 23 (07) ◽  
pp. 1330023 ◽  
Author(s):  
L. ZACHILAS ◽  
M. KATSANIKAS ◽  
P. A. PATSIS
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Fixed Points ◽  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Symplectic Map ◽  
Rotation Method ◽  
Good Agreement

We study the dynamics in the neighborhood of fixed points in a 4D symplectic map by means of the color and rotation method. We compare the results with the corresponding cases encountered in galactic type potentials and we find that they are in good agreement. The fact that the 4D phase space close to fixed points is similar to the 4D representations of the surfaces of section close to periodic orbits, indicates an archetypical 4D pattern for each kind of (in)stability, not only in 3D autonomous Hamiltonian systems with galactic type potentials but for a larger class of dynamical systems. This pattern is successfully visualized with the method we use in the paper.

scholarly journals DYNAMICAL ASPECTS OF LIE-POISSON STRUCTURES

1993 ◽  
Vol 08 (31) ◽  
pp. 2973-2987 ◽  
Author(s):  
F. LIZZI ◽  
G. MARMO ◽  
G. SPARANO ◽  
P. VITALE
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Poisson Bracket ◽  
Lie Groups ◽  
Hamiltonian Systems ◽  
Quantum Groups ◽  
Equations Of Motion ◽  
Poisson Structures ◽  
Dimensional Phase Space ◽  
Deformation Procedure

Quantum groups can be constructed by applying the quantization by deformation procedure to Lie groups endowed with a suitable Poisson bracket. Here we try to develop an understanding of these structures by investigating dynamical systems which are associated with this bracket. We look at SU(2) and SU(1, 1), as submanifolds of a four-dimensional phase space with constraints, and deal with two classes of problems. In the first set of examples we consider some Hamiltonian systems associated with Lie-Poisson structures and we investigate the equations of motion. In the second set of examples we consider systems which preserve the chosen bracket, but are dissipative. However in this approach, they survive the quantization procedure.

Classical invariants for non-Hermitian anharmonic potentials

2020 ◽  
Vol 98 (11) ◽  
pp. 1004-1008
Author(s):  
Ram Mehar Singh ◽  
S.B. Bhardwaj ◽  
Kushal Sharma ◽  
Anand Malik ◽  
Fakir Chand
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Hamiltonian Systems ◽  
One Dimension ◽  
Complex Dynamical Systems ◽  
Complex Phase ◽  
Extended Complex ◽  
Space Approach

Keeping in view the importance of complex dynamical systems, we investigate the classical invariants for some non-Hermitian anharmonic potentials in one dimension. For this purpose, the rationalization method is employed under the elegance of the extended complex phase space approach. The invariants obtained are expected to play an important role in studying complex Hamiltonian systems at the classical as well as quantum levels.

scholarly journals Handling Initial Conditions in Vector Fitting for Real Time Modeling of Power System Dynamics

Energies ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2471
Author(s):  
Tommaso Bradde ◽  
Samuel Chevalier ◽  
Marco De Stefano ◽  
Stefano Grivet-Talocia ◽  
Luca Daniel
Keyword(s):  
Dynamical Systems ◽  
Power System ◽  
System Dynamics ◽  
Real Time ◽  
Initial Conditions ◽  
Test System ◽  
Initial State ◽  
Power System Dynamics ◽  
Vector Fitting ◽  
Time Modeling

This paper develops a predictive modeling algorithm, denoted as Real-Time Vector Fitting (RTVF), which is capable of approximating the real-time linearized dynamics of multi-input multi-output (MIMO) dynamical systems via rational transfer function matrices. Based on a generalization of the well-known Time-Domain Vector Fitting (TDVF) algorithm, RTVF is suitable for online modeling of dynamical systems which experience both initial-state decay contributions in the measured output signals and concurrently active input signals. These adaptations were specifically contrived to meet the needs currently present in the electrical power systems community, where real-time modeling of low frequency power system dynamics is becoming an increasingly coveted tool by power system operators. After introducing and validating the RTVF scheme on synthetic test cases, this paper presents a series of numerical tests on high-order closed-loop generator systems in the IEEE 39-bus test system.

scholarly journals On some kinds of dense orbits in general nonautonomous dynamical systems

2020 ◽  
Vol 7 (1) ◽  
pp. 163-175
Author(s):  
Mehdi Pourbarat
Keyword(s):  
Dynamical Systems ◽  
Initial Conditions ◽  
Point Of View ◽  
Topological Conjugacy ◽  
Topological Transitivity ◽  
Nonautonomous Systems ◽  
Nonautonomous Dynamical Systems ◽  
Sensitive Dependence ◽  
Dense Orbits

AbstractWe study the theory of universality for the nonautonomous dynamical systems from topological point of view related to hypercyclicity. The conditions are provided in a way that Birkhoff transitivity theorem can be extended. In the context of generalized linear nonautonomous systems, we show that either one of the topological transitivity or hypercyclicity give sensitive dependence on initial conditions. Meanwhile, some examples are presented for topological transitivity, hypercyclicity and topological conjugacy.

Pseudorandom Number Generators Based on Chaotic Dynamical Systems

2001 ◽  
Vol 08 (02) ◽  
pp. 137-146 ◽  
Author(s):  
Janusz Szczepański ◽  
Zbigniew Kotulski
Keyword(s):  
Dynamical Systems ◽  
Communication Systems ◽  
Initial Conditions ◽  
Pseudorandom Number ◽  
New Approach ◽  
Chaotic Dynamical Systems ◽  
Pseudorandom Number Generators ◽  
Transmission Of Information ◽  
Extreme Sensitivity ◽  
Contemporary Technology

Pseudorandom number generators are used in many areas of contemporary technology such as modern communication systems and engineering applications. In recent years a new approach to secure transmission of information based on the application of the theory of chaotic dynamical systems has been developed. In this paper we present a method of generating pseudorandom numbers applying discrete chaotic dynamical systems. The idea of construction of chaotic pseudorandom number generators (CPRNG) intrinsically exploits the property of extreme sensitivity of trajectories to small changes of initial conditions, since the generated bits are associated with trajectories in an appropriate way. To ensure good statistical properties of the CPRBG (which determine its quality) we assume that the dynamical systems used are also ergodic or preferably mixing. Finally, since chaotic systems often appear in realistic physical situations, we suggest a physical model of CPRNG.

CHARACTERIZING LOSS OF MEMORY IN CHAOTIC DYNAMICAL SYSTEMS

1991 ◽  
Vol 05 (14) ◽  
pp. 2323-2345 ◽  
Author(s):  
R.E. AMRITKAR ◽  
P.M. GADE
Keyword(s):  
Dynamical Systems ◽  
Initial Conditions ◽  
Chaotic Dynamical Systems

We discuss different methods of characterizing the loss of memory of initial conditions in chaotic dynamical systems.

scholarly journals The Lusternik–Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles

2016 ◽  
Vol 08 (03) ◽  
pp. 545-570 ◽  
Author(s):  
Luca Asselle ◽  
Gabriele Benedetti
Keyword(s):  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Symplectic Form ◽  
Critical Value ◽  
Closed Manifold ◽  
Hamiltonian Flow ◽  
Cotangent Bundles ◽  
Almost All

Let [Formula: see text] be a closed manifold and consider the Hamiltonian flow associated to an autonomous Tonelli Hamiltonian [Formula: see text] and a twisted symplectic form. In this paper we study the existence of contractible periodic orbits for such a flow. Our main result asserts that if [Formula: see text] is not aspherical, then contractible periodic orbits exist for almost all energies above the maximum critical value of [Formula: see text].

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