93 research outputs found
Numerical treatment of oscillary functional differential equations
NOTICE: this is the author’s version of a work that was accepted for publication in Journal of computational and applied mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of computational and applied mathematics, 234(2010), doi: 10.1016/j.cam.2010.01.035This preprint is concerned with oscillatory functional differential equations (that is, those equations where all the solutions oscillate) under a numerical approximation. Our interest is in the preservation of qualitative properties of solutions under a numerical discretisation. We give conditions under which an equation is oscillatory, and consider whether the discrete schemes derived using linear v-methods will also be oscillatory. We conclude with some general theor
Order out of Randomness : Self-Organization Processes in Astrophysics
Self-organization is a property of dissipative nonlinear processes that are
governed by an internal driver and a positive feedback mechanism, which creates
regular geometric and/or temporal patterns and decreases the entropy, in
contrast to random processes. Here we investigate for the first time a
comprehensive number of 16 self-organization processes that operate in
planetary physics, solar physics, stellar physics, galactic physics, and
cosmology. Self-organizing systems create spontaneous {\sl order out of chaos},
during the evolution from an initially disordered system to an ordered
stationary system, via quasi-periodic limit-cycle dynamics, harmonic mechanical
resonances, or gyromagnetic resonances. The internal driver can be gravity,
rotation, thermal pressure, or acceleration of nonthermal particles, while the
positive feedback mechanism is often an instability, such as the
magneto-rotational instability, the Rayleigh-B\'enard convection instability,
turbulence, vortex attraction, magnetic reconnection, plasma condensation, or
loss-cone instability. Physical models of astrophysical self-organization
processes involve hydrodynamic, MHD, and N-body formulations of Lotka-Volterra
equation systems.Comment: 61 pages, 38 Figure
Collocation methods for complex delay models of structured populations
openDottorato di ricerca in Informatica e scienze matematiche e fisicheopenAndo', Alessi
Mathematical frameworks for oscillatory network dynamics in neuroscience
The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear—for example, heteroclinic network attractors. In this review we present a set of mathemat- ical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical frame- work for further successful applications of mathematics to understanding network dynamics in neuroscience
Qualitative analysis of a discrete-time phytoplankton–zooplankton model with Holling type-II response and toxicity
[EN]The interaction among phytoplankton and zooplankton is one of the most important
processes in ecology. Discrete-time mathematical models are commonly used for
describing the dynamical properties of phytoplankton and zooplankton interaction
with nonoverlapping generations. In such type of generations a new age group
swaps the older group after regular intervals of time. Keeping in observation the
dynamical reliability for continuous-time mathematical models, we convert a
continuous-time phytoplankton–zooplankton model into its discrete-time
counterpart by applying a dynamically consistent nonstandard difference scheme.
Moreover, we discuss boundedness conditions for every solution and prove the
existence of a unique positive equilibrium point. We discuss the local stability of
obtained system about all its equilibrium points and show the existence of
Neimark–Sacker bifurcation about unique positive equilibrium under some
mathematical conditions. To control the Neimark–Sacker bifurcation, we apply a
generalized hybrid control technique. For explanation of our theoretical results and to
compare the dynamics of obtained discrete-time model with its continuous
counterpart, we provide some motivating numerical examples. Moreover, from
numerical study we can see that the obtained system and its continuous-time
counterpart are stable for the same values of parameters, and they are unstable for
the same parametric values. Hence the dynamical consistency of our obtained
system can be seen from numerical study. Finally, we compare the modified hybrid
method with old hybrid method at the end of the paper
Dynamics of coupled cell networks: synchrony, heteroclinic cycles and inflation
Copyright © 2011 Springer. The final publication is available at www.springerlink.comWe consider the dynamics of small networks of coupled cells. We usually assume asymmetric inputs and no global or local symmetries in the network and consider equivalence of networks in this setting; that is, when two networks with different architectures give rise to the same set of possible dynamics. Focussing on transitive (strongly connected) networks that have only one type of cell (identical cell networks) we address three questions relating the network structure to dynamics. The first question is how the structure of the network may force the existence of invariant subspaces (synchrony subspaces). The second question is how these invariant subspaces can support robust heteroclinic attractors. Finally, we investigate how the dynamics of coupled cell networks with different structures and numbers of cells can be related; in particular we consider the sets of possible “inflations” of a coupled cell network that are obtained by replacing one cell by many of the same type, in such a way that the original network dynamics is still present within a synchrony subspace. We illustrate the results with a number of examples of networks of up to six cells
Dynamical Models of Biology and Medicine
Mathematical and computational modeling approaches in biological and medical research are experiencing rapid growth globally. This Special Issue Book intends to scratch the surface of this exciting phenomenon. The subject areas covered involve general mathematical methods and their applications in biology and medicine, with an emphasis on work related to mathematical and computational modeling of the complex dynamics observed in biological and medical research. Fourteen rigorously reviewed papers were included in this Special Issue. These papers cover several timely topics relating to classical population biology, fundamental biology, and modern medicine. While the authors of these papers dealt with very different modeling questions, they were all motivated by specific applications in biology and medicine and employed innovative mathematical and computational methods to study the complex dynamics of their models. We hope that these papers detail case studies that will inspire many additional mathematical modeling efforts in biology and medicin
Nonlinear and evolutionary phenomena in deterministic growing economies
We discuss the implications of nonlinearity in competitive models of optimal
endogenous growth. Departing from a simple representative agent setup with
convex risk premium and investment adjustment costs, we define an open economy
dynamic optimization problem and show that the optimal control solution is given
by an autonomous nonlinear vector field in <3 with multiple equilibria and no optimal
stable solutions. We give a thorough analytical and numerical analysis of this
system qualitative dynamics and show the existence of local singularities, such as
fold (saddle-node), Hopf and Fold-Hopf bifurcations of equilibria. Finally, we discuss
the policy implications of global nonlinear phenomena. We focus on dynamic
scenarios arising in the vicinity of Fold-Hopf bifurcations and demonstrate the existence
of global dynamic phenomena arising from the complex organization of the
invariant manifolds of this system. We then consider this setup in a non-cooperative
differential game environment, where asymmetric players choose open loop no feedback
strategies and dynamics are coupled by an aggregate risk premium mechanism.
When only convex risk premium is considered, we show that these games have a
specific state-separability property, where players have optimal, but naive, beliefs
about the evolution of the state of the game. We argue that the existence of optimal
beliefs in this fashion, provides a unique framework to study the implications
of the self-confirming equilibrium (SCE) hypothesis in a dynamic game setup. We
propose to answer the following question. Are players able to concur on a SCE,
where their expectations are self-fulfilling? To evaluate this hypothesis we consider
a simple conjecture. If beliefs bound the state-space of the game asymptotically
and strategies are Lipschitz continuous, then it is possible to describe SCE solutions
and evaluate the qualitative properties of equilibrium. If strategies are not smooth,
which is likely in environments where belief-based solutions require players to learn
a SCE, then asymptotic dynamics can be evaluated numerically as a Hidden Markov
Model (HMM). We discuss this topic for a class of games where players lack the
relevant information to pursue their optimal strategies and have to base their decisions
on subjective beliefs. We set up one of the games proposed as a multi-objective
optimization problem under uncertainty and evaluate its asymptotic solution as a
multi-criteria HMM.We show that under a simple linear learning regime there is convergence
to a SCE and portray strong emergence phenomena as a result of persistent
uncertainty
Construction and analysis of efficient numerical methods to solve mathematical models of TB and HIV co-infection
Philosophiae Doctor - PhDThe global impact of the converging dual epidemics of tuberculosis (TB) and human immunodeficiency virus (HIV) is one of the major public health challenges of our time, because in many countries, human immunodeficiency virus (HIV) and mycobacterium tuberculosis (TB) are among the leading causes of morbidity and mortality. It is found that infection with HIV increases the risk of reactivating latent TB infection, and HIV-infected individuals who acquire new TB infections have high rates of disease progression. Research has shown that these two diseases are enormous public health burden, and unfortunately, not much has been done in terms of modeling the dynamics of HIV-TB co-infection at a population level. In this thesis, we study these models and design and analyze robust numerical methods to solve them. To proceed in this direction, first we study the sub-models and then the full model. The first sub-model describes the transmission dynamics of HIV that accounts for behavior change. The impact of HIV educational campaigns is also studied. Further, we explore the effects of behavior change and different responses of individuals to educational campaigns in a situation where individuals may not react immediately to these campaigns. This is done by considering a distributed time delay in the HIV sub-model. This leads to Hopf bifurcations around the endemic equilibria of the model. These bifurcations correspond to the existence of periodic solutions that oscillate around the equilibria at given thresholds. Further, we show how the delay can result in more HIV infections causing more increase in the HIV prevalence. Part of this study is then extended to study a co-infection model of HIV-TB. A thorough bifurcation analysis is carried out for this model. Robust numerical methods are then designed and analyzed for these models. Comparative numerical results are also provided for each model.South Afric
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