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Lecture

Dynamic Systems: Course Review

In course

BIO-341: Dynamical systems in biology

Life is non-linear. This course introduces dynamical systems as a technique for modelling simple biological processes. The emphasis is on the qualitative and numerical analysis of non-linear dynamical models. Examples are taken from biology and population ...

Description

This lecture covers the review of dynamic systems, trajectories, and growth models for populations. It discusses the stability of fixed points and linearization of models, with a focus on the logistic equation and its approximations.

Instructors (2)

Felix Naef

Felix Naef studied theoretical physics at the ETHZ and obtained his PhD from the EPFL in 2000. He then received postdoctoral training at the Center for Studies in Physics and Biology at the Rockefeller University (NYC) under the guidance of Prof. Magnasco. His research focuses on the modeling and interpretation of high-throughput functional data and the study of biomolecular oscillators. He joined ISREC as an associate scientist in early 2004 and is currently Associate Professor in the Institute of Bioengineering (IBI).

Julian Charles Shillcock

Julian received his PhD at Simon Fraser University in Canada for work on Monte Carlo simulations of liquid crystal phase transitions and the elastic properties of fluid and polymerized membranes. He was a Group Leader at the Max Planck Institute of Colloids and Interfaces, Germany for five years applying coarse-grained simulation techniques - principally Dissipative Particle Dynamics (DPD) and Brownian Dynamics - to equilibrium and dynamic properties of fluid lipid membranes. A major target of this research was to reveal the molecular rearrangements that occur during vesicle fusion. During this time, he developed a commercially-available, parallel DPD code that is being used by several universities. He was an Associate Professor at MEMPHYS in the Department of Physics and Chemistry, University of Southern Denmark. He has also worked in industry, designing and writing software for communication satellite simulations (British Aerospace, 1986-1990), and fluid simulation software (Accelrys, Inc., 1998-1999). He joined the Blue Brain Project in 2011, and uses mesoscale simulation techniques, together with theoretical analysis and collaborations with experimentalists, to study the dynamics of cellular processes. Current projects include simulating the formation of the post-synaptic density in dendritic spines and developing theoretical models of the supramolecular organisation of synapses. He also teaches Master's and PhD courses in computational cell biology and biophysics. He wrote and maintains an open source dissipative particle dynamics simulation code: https://github.com/Osprey-DPD/osprey-dpd Orcid: orcid.org/0000-0002-7885-735X

Official source

Ontological neighbourhood

Systems science and engineering

Systems theory: Chaos theory

Related concepts (32)

Fixed point (mathematics)

hatnote|1=Fixed points in mathematics are not to be confused with other uses of "fixed point", or stationary points where math|1=f(x) = 0. In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation. Specifically for functions, a fixed point is an element that is mapped to itself by the function. Formally, c is a fixed point of a function f if c belongs to both the domain and the codomain of f, and f(c) = c.

Dynamical system

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured.

Fixed-point iteration

In numerical analysis, fixed-point iteration is a method of computing fixed points of a function. More specifically, given a function defined on the real numbers with real values and given a point in the domain of , the fixed-point iteration is which gives rise to the sequence of iterated function applications which is hoped to converge to a point . If is continuous, then one can prove that the obtained is a fixed point of , i.e., More generally, the function can be defined on any metric space with values in that same space.

Fixed-point theorem

In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point.

Dynamical systems theory

Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle.

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Covers the formalism and bases of dynamic systems, including differential equations and non-linear systems.

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