Generation of nonlinear ordinary differential equations with given Lie algebra of point symmetries. (Erzeugung nichtlinearer gewöhnlicher Differentialgleichungen mit vorgegebener Lie-Algebra von Punktsymmetrien.)
Generic bifurcation of reversible vector fields on a 2-bidimensional manifold.
In this paper we deal with reversible vector fields on a 2-dimensional manifold having a codimension one submanifold as its symmetry axis. We classify generically the one parameter families of such vector fields. As a matter of fact, aspects of structural stability and codimension one bifurcation are analysed.
Geometric linearization of ordinary differential equations.
Geometry of third order ODE systems
We compute cohomology spaces of Lie algebras that describe differential invariants of third order ordinary differential equations. We prove that the algebra of all differential invariants is generated by 2 tensorial invariants of order 2, one invariant of order 3 and one invariant of order 4. The main computational tool is a Serre-Hochschild spectral sequence and the representation theory of semisimple Lie algebras. We compute differential invariants up to degree 2 as application.
Homogeneous systems of higher-order ordinary differential equations
The concept of homogeneity, which picks out sprays from the general run of systems of second-order ordinary differential equations in the geometrical theory of such equations, is generalized so as to apply to equations of higher order. Certain properties of the geometric concomitants of a spray are shown to continue to hold for higher-order systems. Third-order equations play a special role, because a strong form of homogeneity may apply to them. The key example of a single third-order equation...
Homogeneous Systems with a Quiescent Phase
Recently the effect of a quiescent phase (or dormant/resting phase in applications) on the dynamics of a system of differential equations has been investigated, in particular with respect to stability properties of stationary points. It has been shown that there is a general phenomenon of stabilization against oscillations which can be cast in rigorous form. Here we investigate, for homogeneous systems, the effect of a quiescent phase, and more generally, a phase with slower dynamics. We show that...
Impulsive differential inclusions with constraints.
Input-output systems in Biology and Chemistry and a class of mathematical models describing them
Our aim is to show a class of mathematical models in application to some problems of cell biology. Typically, our models are described via classical chemical networks. This property is visualized by a conservation law. Mathematically, this conservation law guarantees most of the mathematical properties of the models such as global existence and uniqueness of solutions as well as positivity of the solutions for positive data. These properties are consequences of the fact that the infinitesimal generators...
Invariants of kinetic differential equations.
Lie systems: theory, generalisations, and applications [Book]
Linear ODEs and -modules, solving and decomposing equations using symmetry methods.
Local integrating factors.
Multi-parameter symmetries of first order ordinary differential equations.
Noether symmetries and critical exponents.
Nonlinear dynamical systems and KCC-theory.
On computer-aided solving differential equations and stability study of markets.
On plane polynomial vector fields and the Poincaré problem.
On the Cordovan ogive
On the integrability of quasihomogeneous systems and quasidegenerate infinity systems.
On the linearization of second-order differential and difference equations.