Chiara Mocennihttp://www.dii.unisi.it/~mocenni

List of projects

Chemical reactionsOscillations can arise in a macroscopic medium if the system is sufficiently far from the state of thermodynamic equilibrium (Nicolis and Prigogine, 1977).

stirred unstirred

Brusselator

Il modello Lotka-Volterra

x = prey y = predator(x) = growth of prey = mortality of predatorp(x) = functional response of the predator

Ecological systems

Economic systemsCournot competition is an economic model used to describe an industry structure in which companies compete on the amount of output they will produce, which they decide on independently of each other and at the same time. It is named after Antoine Augustin Cournot (1801–1877). It has the following features:

There is more than one firm and all firms produce a homogeneous product, i.e. there is no product differentiationFirms do not cooperate, i.e. there is no collusionFirms have market power, i.e. each firm's output decision affects the good's priceThe number of firms is fixedFirms compete in quantities, and choose quantities simultaneouslyThe firms are economically rational and act strategically, usually seeking to maximize profit given their competitors' decisionsDiscrete time or continuous time nonlinear equations

Epidemic modeling

• http://nrich.maths.org/4489

W. O. Kermack and A. G. McKendrick model

S: population of suceptiblesI: population if infectedR: population of recovered

Time series analysis

• Suppose that you only have the recordings of some measures correlated to a state variable of a dynamical system

• You can estimate the level of chaoticity in the time series by calculatingsome indicators, such as the Lyapunov exponent

• You can reconstruct approximately the original system by using delayed coordinates

FractalsThe Mandelbrot set is the set of values of c in the complex plane for which the orbit of 0 under iteration of the complex quadratic polynomial

zn+1 = zn2 + c

remains bounded. That is, a complex number c is part of the Mandelbrot set if, when starting with z0 = 0 and applying the iteration repeatedly, the absolute value of zn remains bounded however large n gets.

What about iterating the logistic map instead?

Matlab Toolbox

• Development of some Matlab tools for the analysis of the bifurcations of nonlinear systems

Billiard tableA billiard is a dynamical system in which a particle alternates between motion in a straight line and specular reflections from a boundary. When the particle hits the boundary it reflects from it without loss of speed.

What about simulating and calculating the Lyapunov exponent of the trajectories?

Ott, Grebogi & Yorke

..It is shown that one can convert a chaotic attractor to any one of a large number of possible attracting time-periodic motions by making only small time-dependent perturbations of an available system parameter…

Chaos control

Complexity science: The elephant in the dark

Complexity science

Complex Networks

dove

time derivative

lapplacian (diffusion)

u(x,y,t) concentration of a species,

D diffusion coefficient,

f(u,x,y,t) reaction term

Reaction-diffusion systems

Other applications

• Brain dynamics• Double pendulum• .• .• .