The Poincaré-Cartan forms of one-dimensional variational integrals

2020 ◽  
Vol 70 (6) ◽  
pp. 1381-1412
Author(s):  
Veronika Chrastinová ◽  
Václav Tryhuk
Keyword(s):  
Differential Equations ◽  
Variational Problem ◽  
Ordinary Differential Equations ◽  
Differential Forms ◽  
Point Of View ◽  
Full Generality ◽  
One Dimensional ◽  
Variational Integrals

AbstractFundamental concepts for variational integrals evaluated on the solutions of a system of ordinary differential equations are revised. The variations, stationarity, extremals and especially the Poincaré-Cartan differential forms are relieved of all additional structures and subject to the equivalences and symmetries in the widest possible sense. Theory of the classical Lagrange variational problem eventually appears in full generality. It is presented from the differential forms point of view and does not require any intricate geometry.

scholarly journals Automorphisms of Ordinary Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-32 ◽  
Author(s):  
Václav Tryhuk ◽  
Veronika Chrastinová
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Local Theory ◽  
Full Generality ◽  
One Dimensional ◽  
Underdetermined Systems ◽  
Dependent Variables ◽  
Variational Integrals ◽  
Independent Variable ◽  
Internal Symmetries

The paper deals with the local theory of internal symmetries of underdetermined systems of ordinary differential equations in full generality. The symmetries need not preserve the choice of the independent variable, the hierarchy of dependent variables, and the order of derivatives. Internal approach to the symmetries of one-dimensional constrained variational integrals is moreover proposed without the use of multipliers.

scholarly journals On the internal approach to differential equations. 1. The involutiveness and standard basis

2016 ◽  
Vol 66 (4) ◽  
Author(s):  
Veronika Chrastinová ◽  
Václav Tryhuk
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Commutative Algebra ◽  
Geometrical Theory ◽  
Technical Tool ◽  
Full Generality ◽  
Dependent Variables ◽  
Variational Integrals ◽  
Independent Variable

AbstractThe article treats the geometrical theory of partial differential equations in the absolute sense, i.e., without any additional structures and especially without any preferred choice of independent and dependent variables. The equations are subject to arbitrary transformations of variables in the widest possible sense. In this preparatory Part 1, the involutivity and the related standard bases are investigated as a technical tool within the framework of commutative algebra. The particular case of ordinary differential equations is briefly mentioned in order to demonstrate the strength of this approach in the study of the structure, symmetries and constrained variational integrals under the simplifying condition of one independent variable. In full generality, these topics will be investigated in subsequent Parts of this article.

scholarly journals Symmetries and Conservation Laws for Some Compacton Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
M. L. Gandarias ◽  
M. S. Bruzón ◽  
M. Rosa
Keyword(s):  
Differential Equations ◽  
Qualitative Analysis ◽  
Conservation Laws ◽  
Ordinary Differential Equations ◽  
Dynamical Behaviour ◽  
Point Of View ◽  
Nonlinear Dispersion ◽  
Multipliers Method

We consider some equations with compacton solutions and nonlinear dispersion from the point of view of Lie classical reductions. The reduced ordinary differential equations are suitable for qualitative analysis and their dynamical behaviour is described. We derive by using the multipliers method some nontrivial conservation laws for these equations.

Qualitative properties of nonlinear ordinary differential equations

1977 ◽  
Vol 79 (1-2) ◽  
pp. 79-85 ◽  
Author(s):  
Nelson Onuchic ◽  
Plácido Z. Táboas
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Ordinary Differential Equation ◽  
Point Of View ◽  
Nonlinear Ordinary Differential Equations ◽  
Qualitative Properties ◽  
Image Position

SynopsisThe perturbed linear ordinary differential equationis considered. Adopting the same approach of Massera and Schäffer [6], Corduneanu states in [2] the existence of a set of solutions of (1) contained in a given Banach space. In this paper we investigate some topological aspects of the set and analyze some of the implications from a point of view ofstability theory.

scholarly journals Quotients of Euler Equations on Space Curves

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 186
Author(s):  
Anna Duyunova ◽  
Valentin Lychagin ◽  
Sergey Tychkov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Euler Equations ◽  
Gas Flow ◽  
Euler System ◽  
Partial Differential ◽  
One Dimensional ◽  
Space Curves ◽  
Ode System

Quotients of partial differential equations are discussed. The quotient equation for the Euler system describing a one-dimensional gas flow on a space curve is found. An example of using the quotient to solve the Euler system is given. Using virial expansion of the Planck potential, we reduce the quotient equation to a series of systems of ordinary differential equations (ODEs). Possible solutions of the ODE system are discussed.

scholarly journals Chaos in an anharmonic oscillator

1995 ◽  
Vol 37 (2) ◽  
pp. 186-207 ◽  
Author(s):  
J. R. Christie ◽  
K. Gopalsamy
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Autonomous System ◽  
Anharmonic Oscillator ◽  
Homoclinic Orbits ◽  
Chaotic Behaviour ◽  
Differential Systems ◽  
Melnikov's Method ◽  
One Dimensional ◽  
Melnikov’S Method

AbstractUsing Melnikov's method, the existence of chaotic behaviour in the sense of Smale in a particular time-periodically perturbed planar autonomous system of ordinary differential equations is established. Examples of planar autonomous differential systems with homoclinic orbits are provided, and an application to the dynamics of a one-dimensional anharmonic oscillator is given.

CLUSTERING AND SYNCHRONIZATION IN A ONE-DIMENSIONAL MODEL FOR THE CA3 REGION OF THE HIPPOCAMPUS

2002 ◽  
Vol 12 (11) ◽  
pp. 2641-2653 ◽  
Author(s):  
N. MONTEJO ◽  
M. N. LORENZO ◽  
V. PÉREZ-MUÑUZURI ◽  
V. PÉREZ-VILLAR
Keyword(s):  
Time Delay ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Spatiotemporal Patterns ◽  
Transmission Time ◽  
Dimensional Model ◽  
One Dimensional ◽  
Dimensional Array ◽  
Ca3 Region

The behavior of a system of coupled ordinary differential equations is studied in order to characterize the CA3 region of the hippocampus. Clustering and synchronization behavior in a one-dimensional array of cells modeled by a modified Morris–Lecar model is analyzed in terms of a time delay included in the model. The random formation of phase dislocations whose number increases with the time delay seems to be responsible for complex spatiotemporal patterns that have been observed. Alterations to the transmission time between cells have been simulated by adding some noise to the system.

scholarly journals Ordinary differential equations for the dynamic characteristics of heating boilers

2018 ◽  
Vol 194 ◽  
pp. 01024
Author(s):  
Sergei Khaustov ◽  
Olga Guk ◽  
Igor Razov
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Control Systems ◽  
Dynamic Characteristics ◽  
Solid Fuel ◽  
Computational Time ◽  
One Dimensional ◽  
Automatic Control Systems ◽  
Set Up

The paper presents ordinary differential equations for the dynamic characteristics of a solid fuel boiler that can be combined into one-dimensional nonstationary mathematical model for simulating long-term dynamics of a solid fuel boiler. This model requires less computational time for a qualitative simulation of boiler’s operation than known CFD-solutions. The long-term dynamic model presented can help to determine annual costs, to set up automatic control systems and to detect dangerous deviations in the project.

scholarly journals The effect of domain growth on spatial correlations

2016 ◽  
Author(s):  
Robert JH Ross ◽  
Christian A Yates ◽  
Ruth E Baker
Keyword(s):  
Differential Equations ◽  
Cell Motility ◽  
Ordinary Differential Equations ◽  
Cell Movement ◽  
Domain Growth ◽  
Continuum Approximation ◽  
Spatial Correlations ◽  
One Dimensional ◽  
Growing Domain ◽  
Pair Density

Mathematical models describing cell movement and proliferation are important research tools for the understanding of many biological processes. In this work we present methods to include the effects of domain growth on the evolution of spatial correlations between agent locations in a continuum approximation of a one-dimensional lattice-based model of cell motility and proliferation. This is important as the inclusion of spatial correlations in continuum models of cell motility and proliferation without domain growth has previously been shown to be essential for their accuracy in certain scenarios. We include the effect of spatial correlations by deriving a system of ordinary differential equations that describe the expected evolution of individual and pair density functions for agents on a growing domain. We then demonstrate how to simplify this system of ordinary differential equations by using an appropriate approximation. This simplification allows domain growth to be included in models describing the evolution of spatial correlations between agents in a tractable manner.

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