778 research outputs found
Reduction of dimension for nonlinear dynamical systems
We consider reduction of dimension for nonlinear dynamical systems. We
demonstrate that in some cases, one can reduce a nonlinear system of equations
into a single equation for one of the state variables, and this can be useful
for computing the solution when using a variety of analytical approaches. In
the case where this reduction is possible, we employ differential elimination
to obtain the reduced system. While analytical, the approach is algorithmic,
and is implemented in symbolic software such as {\sc MAPLE} or {\sc SageMath}.
In other cases, the reduction cannot be performed strictly in terms of
differential operators, and one obtains integro-differential operators, which
may still be useful. In either case, one can use the reduced equation to both
approximate solutions for the state variables and perform chaos diagnostics
more efficiently than could be done for the original higher-dimensional system,
as well as to construct Lyapunov functions which help in the large-time study
of the state variables. A number of chaotic and hyperchaotic dynamical systems
are used as examples in order to motivate the approach.Comment: 16 pages, no figure
Thomas decompositions of parametric nonlinear control systems
This paper presents an algorithmic method to study structural properties of
nonlinear control systems in dependence of parameters. The result consists of a
description of parameter configurations which cause different control-theoretic
behaviour of the system (in terms of observability, flatness, etc.). The
constructive symbolic method is based on the differential Thomas decomposition
into disjoint simple systems, in particular its elimination properties
On the General Analytical Solution of the Kinematic Cosserat Equations
Based on a Lie symmetry analysis, we construct a closed form solution to the
kinematic part of the (partial differential) Cosserat equations describing the
mechanical behavior of elastic rods. The solution depends on two arbitrary
analytical vector functions and is analytical everywhere except a certain
domain of the independent variables in which one of the arbitrary vector
functions satisfies a simple explicitly given algebraic relation. As our main
theoretical result, in addition to the construction of the solution, we proof
its generality. Based on this observation, a hybrid semi-analytical solver for
highly viscous two-way coupled fluid-rod problems is developed which allows for
the interactive high-fidelity simulations of flagellated microswimmers as a
result of a substantial reduction of the numerical stiffness.Comment: 14 pages, 3 figure
Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)
Oxford, UK, 26 August 200
- …