420 research outputs found
Finite-time Lagrangian transport analysis: Stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents
We consider issues associated with the Lagrangian characterisation of flow
structures arising in aperiodically time-dependent vector fields that are only
known on a finite time interval. A major motivation for the consideration of
this problem arises from the desire to study transport and mixing problems in
geophysical flows where the flow is obtained from a numerical solution, on a
finite space-time grid, of an appropriate partial differential equation model
for the velocity field. Of particular interest is the characterisation,
location, and evolution of "transport barriers" in the flow, i.e. material
curves and surfaces. We argue that a general theory of Lagrangian transport has
to account for the effects of transient flow phenomena which are not captured
by the infinite-time notions of hyperbolicity even for flows defined for all
time. Notions of finite-time hyperbolic trajectories, their finite time stable
and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields
and associated Lagrangian coherent structures have been the main tools for
characterizing transport barriers in the time-aperiodic situation. In this
paper we consider a variety of examples, some with explicit solutions, that
illustrate, in a concrete manner, the issues and phenomena that arise in the
setting of finite-time dynamical systems. Of particular significance for
geophysical applications is the notion of "flow transition" which occurs when
finite-time hyperbolicity is lost, or gained. The phenomena discovered and
analysed in our examples point the way to a variety of directions for rigorous
mathematical research in this rapidly developing, and important, new area of
dynamical systems theory
Efficient method for detection of periodic orbits in chaotic maps and flows
An algorithm for detecting unstable periodic orbits in chaotic systems [Phys.
Rev. E, 60 (1999), pp. 6172-6175] which combines the set of stabilising
transformations proposed by Schmelcher and Diakonos [Phys. Rev. Lett., 78
(1997), pp. 4733-4736] with a modified semi-implicit Euler iterative scheme and
seeding with periodic orbits of neighbouring periods, has been shown to be
highly efficient when applied to low-dimensional system. The difficulty in
applying the algorithm to higher dimensional systems is mainly due to the fact
that the number of stabilising transformations grows extremely fast with
increasing system dimension. In this thesis, we construct stabilising
transformations based on the knowledge of the stability matrices of already
detected periodic orbits (used as seeds). The advantage of our approach is in a
substantial reduction of the number of transformations, which increases the
efficiency of the detection algorithm, especially in the case of
high-dimensional systems. The performance of the new approach is illustrated by
its application to the four-dimensional kicked double rotor map, a
six-dimensional system of three coupled H\'enon maps and to the
Kuramoto-Sivashinsky system in the weakly turbulent regime.Comment: PhD thesis, 119 pages. Due to restrictions on the size of files
uploaded, some of the figures are of rather poor quality. If necessary a
quality copy may be obtained (approximately 1MB in pdf) by emailing me at
[email protected]
Global modes and nonlinear analysis of inverted-flag flapping
An inverted flag has its trailing edge clamped and exhibits dynamics distinct
from that of a conventional flag, whose leading edge is restrained. We perform
nonlinear simulations and a global stability analysis of the inverted-flag
system for a range of Reynolds numbers, flag masses and stiffnesses. Our global
stability analysis is based on a linearisation of the fully-coupled
fluid-structure system of equations. The calculated equilibria are steady-state
solutions of the fully-coupled nonlinear equations. By implementing this
approach, we (i) explore the mechanisms that initiate flapping, (ii) study the
role of vortex shedding and vortex-induced vibration (VIV) in large-amplitude
flapping, and (iii) characterise the chaotic flapping regime. For point (i), we
identify a deformed-equilibrium state and show through a global stability
analysis that the onset of flapping is due to a supercritical Hopf bifurcation.
For large-amplitude flapping, point (ii), we confirm the arguments of Sader et
al. (2016) that for a range of parameters this regime is a VIV. We also show
that there are other flow regimes for which large-amplitude flapping persists
and is not a VIV. Specifically, flapping can occur at low Reynolds numbers
(), albeit via a previously unexplored mechanism. Finally, with respect to
point (iii), chaotic flapping has been observed experimentally for Reynolds
numbers of , and here we show that chaos also persists at a moderate
Reynolds number of 200. We characterise this chaotic regime and calculate its
strange attractor, whose structure is controlled by the above-mentioned
deformed equilibria and is similar to a Lorenz attractor. These results are
contextualised with bifurcation diagrams that depict the different equilibria
and various flapping regimes
On Dynamics and Invariant Sets in Predator-Prey Maps
A multitude of physical, chemical, or biological systems evolving in discrete time can be modelled and studied using difference equations (or iterative maps). Here we discuss local and global dynamics for a predator-prey two-dimensional map. The system displays an enormous richness of dynamics including extinctions, co-extinctions, and both ordered and chaotic coexistence. Interestingly, for some regions we have found the so-called hyperchaos, here given by two positive Lyapunov exponents. An important feature of biological dynamical systems, especially in discrete time, is to know where the dynamics lives and asymptotically remains within the phase space, that is, which is the invariant set and how it evolves under parameter changes. We found that the invariant set for the predator-prey map is very sensitive to parameters, involving the presence of escaping regions for which the orbits go out of the domain of the system (the species overcome the carrying capacity) and then go to extinction in a very fast manner. This theoretical finding suggests a potential dynamical fragility by which unexpected and sharp extinctions may take place
On the solution of the Riccati differential equation arising from the LQ optimal control problem
In this paper we consider the matrix Riccati differential equation (RDE) that arises from linear-quadratic (LQ) optimal control problems. In particular, we establish explicit closed formulae for the solution of the RDE with a terminal condition using particular solutions of the associated algebraic Riccati equation. We discuss how these formulae change as assumptions are progressively weakened. An application to LQ optimal control is briefly analysed
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