31 research outputs found
A graph-theoretic condition for irreducibility of a set of cone preserving matrices
Given a closed, convex and pointed cone K in R^n , we present a result which infers K-irreducibility of sets of K-quasipositive matrices from strong connectedness of certain bipartite digraphs. The matrix-sets are defined via products, and the main result is relevant to applications in biology and chemistry. Several examples are presented
Directed transport of two interacting particles in a washboard potential
We study the conservative and deterministic dynamics of two nonlinearly
interacting particles evolving in a one-dimensional spatially periodic
washboard potential. A weak tilt of the washboard potential is applied biasing
one direction for particle transport. However, the tilt vanishes asymptotically
in the direction of bias. Moreover, the total energy content is not enough for
both particles to be able to escape simultaneously from an initial potential
well; to achieve transport the coupled particles need to interact
cooperatively. For low coupling strength the two particles remain trapped
inside the starting potential well permanently. For increased coupling strength
there exists a regime in which one of the particles transfers the majority of
its energy to the other one, as a consequence of which the latter escapes from
the potential well and the bond between them breaks. Finally, for suitably
large couplings, coordinated energy exchange between the particles allows them
to achieve escapes -- one particle followed by the other -- from consecutive
potential wells resulting in directed collective motion. The key mechanism of
transport rectification is based on the asymptotically vanishing tilt causing a
symmetry breaking of the non-chaotic fraction of the dynamics in the mixed
phase space. That is, after a chaotic transient, only at one of the boundaries
of the chaotic layer do resonance islands appear. The settling of trajectories
in the ballistic channels associated with transporting islands provides
long-range directed transport dynamics of the escaping dimer
Emergence of continual directed flow in Hamiltonian systems
We propose a minimal model for the emergence of a directed flow in autonomous
Hamiltonian systems. It is shown that internal breaking of the spatio-temporal
symmetries, via localised initial conditions, that are unbiased with respect to
the transporting degree of freedom, and transient chaos conspire to form the
physical mechanism for the occurrence of a current. Most importantly, after
passage through the transient chaos, trajectories perform solely regular
transporting motion so that the resulting current is of continual ballistic
nature. This has to be distinguished from the features of transport reported
previously for driven Hamiltonian systems with mixed phase space where
transport is determined by intermittent behaviour exhibiting power-law decay
statistics of the duration of regular ballistic periods
Nonlinear response of a linear chain to weak driving
We study the escape of a chain of coupled units over the barrier of a
metastable potential. It is demonstrated that a very weak external driving
field with suitably chosen frequency suffices to accomplish speedy escape. The
latter requires the passage through a transition state the formation of which
is triggered by permanent feeding of energy from a phonon background into humps
of localised energy and elastic interaction of the arising breather solutions.
In fact, cooperativity between the units of the chain entailing coordinated
energy transfer is shown to be crucial for enhancing the rate of escape in an
extremely effective and low-energy cost way where the effect of entropic
localisation and breather coalescence conspire
A computational procedure to detect a new type of high-dimensional chaotic saddle and its application to the 3D Hill’s problem
From collective periodic running states to completely chaotic synchronised states in coupled particle dynamics
We consider the damped and driven dynamics of two interacting particles
evolving in a symmetric and spatially periodic potential. The latter is exerted
to a time-periodic modulation of its inclination. Our interest is twofold:
Firstly we deal with the issue of chaotic motion in the higher-dimensional
phase space. To this end a homoclinic Melnikov analysis is utilised assuring
the presence of transverse homoclinic orbits and homoclinic bifurcations for
weak coupling allowing also for the emergence of hyperchaos. In contrast, we
also prove that the time evolution of the two coupled particles attains a
completely synchronised (chaotic) state for strong enough coupling between
them. The resulting `freezing of dimensionality' rules out the occurrence of
hyperchaos. Secondly we address coherent collective particle transport provided
by regular periodic motion. A subharmonic Melnikov analysis is utilised to
investigate persistence of periodic orbits. For directed particle transport
mediated by rotating periodic motion we present exact results regarding the
collective character of the running solutions entailing the emergence of a
current. We show that coordinated energy exchange between the particles takes
place in such a manner that they are enabled to overcome - one particle
followed by the other - consecutive barriers of the periodic potential
resulting in collective directed motion
A hybrid spatiotemporal model of PCa dynamics and insights into optimal therapeutic strategies
Using a hybrid cellular automaton with stochastic elements, we investigate the effectiveness of multiple drug therapies on prostate cancer (PCa) growth. The ability of Androgen Deprivation Therapy to reduce PCa growth represents a milestone in prostate cancer treatment, nonetheless most patients eventually become refractory and develop castration-resistant prostate cancer. In recent years, a “second generation” drug called enzalutamide has been used to treat advanced PCa, or patients already exposed to chemotherapy that stopped responding to it. However, tumour resistance to enzalutamide is not well understood, and in this context, preclinical models and in silico experiments (numerical simulations) are key to understanding the mechanisms of resistance and to assessing therapeutic settings that may delay or prevent the onset of resistance. In our mathematical system, we incorporate cell phenotype switching to model the development of increased drug resistance, and consider the effect of the micro-environment dynamics on necrosis and apoptosis of the tumour cells. The therapeutic strategies that we explore include using a single drug (enzalutamide), and drug combinations (enzalutamide and everolimus or cabazitaxel) with different treatment schedules. Our results highlight the effectiveness of alternating therapies, especially alternating enzalutamide and cabazitaxel over a year, and a comparison is made with data taken from TRAMP mice to verify our findings
Rigorous computer-assisted bounds on the period doubling renormalisation fixed point and eigenfunctions in maps with critical point of degree 4
We gain tight rigorous bounds on the renormalisation fixed point for period
doubling in families of unimodal maps with degree critical point. We use a
contraction mapping argument to bound essential eigenfunctions and eigenvalues
for the linearisation of the operator and for the operator controlling the
scaling of added noise. Multi-precision arithmetic with rigorous directed
rounding is used to bound operations in a space of analytic functions yielding
tight bounds on power series coefficients and universal constants to over
significant figures.Comment: 15 pages, 8 figure