The balance between injection and dissipation of a physical 
quantity leads a system to behave in a complicated manner, 
producing a rich variety of spatiotemporal structures. 
Commonly, systems far from equilibrium are described by 
nonlinear differential equations (NDEs) or coupled maps. In 
general, it is not possible to obtain analytic solutions 
for these NDEs.
Nevertheless, in the last decades an extensive variety of 
general techniques have been developed to obtain 
approximate solutions close to the onset of the 
instabilities. In a determinist system, different type of 
states can be found varying the parameters and the initial 
conditions. By changing one parameter in the system, it can 
pass from one state to another. For instance, in an 
extended system a homogeneous state can become a pattern. 
This type of bifurcation is called spatial instability.
The pattern can be regular or chaotic. The latter term 
means that pattern’s behavior in a long time window is 
aperiodic and sensible to the initial conditions. One type 
of physical systems that presents a variety of complex 
phenomena is the magnetic systems, especially when a 
parametric driven forcing is acting over the system. In 
fact, in low dimensional magnetic system chaotic behaviors 
may appear when the magnetic field is a time dependent 
function. On the hand, extended systems can exhibit 
dissipative Solitons or Faraday waves depending on the 
amplitude and frequency of the driven forcing.
The course pretends to cover problems concerning systems 
far from equilibrium where there is a competition
between injection and dissipation of energy. The prototypes 
models will be based on contemporary magnetic systems at
nanometer scales. Theoretical and numerical approaches will 
be treated. These problems are interesting from both 
physical and mathematical point of view. The first goal of 
the course is to motivate students to solve complex 
problems. The course will focus on different aspect of 
parametric instabilities in magnetic systems, at zero-, 
one- and two- spatial dimensions. The different types of 
dimensions have a practical significance because they can 
be used to model particles, wires, stripes as well as
spin-valve oscillator devices, which are central part of 
nanoscience and have potential technological applications. 
The second goal is to show the intrinsic correspondence 
between the physics and mathematics of the magnetic 
systems. The derivation of the spatiotemporal evolution 
equations will be performed and some features of them will 
be analyzed. The final goal is to show how several branches 
of mathematics are indispensable to characterize the 
dynamical behavior of such systems. In particular,
different methods based on bifurcation analysis, 
perturbation methods, and the normal form theory will be 
introduced. In addition, numerical simulations techniques 
will be examined. In particular, the Lyapunov exponents 
method will be shown.
The course will benefit the students of graduate and post 
graduate levels, and academicians of mathematics and
physics to acquire a new experience to apply mathematical 
methods in modern physical problems.
Modules Foundations of magnetism, Magnetization dynamics , 
Non-autonomous dynamics , Simulations
Conservative systems, 1D, 2D systems out of equilibrium, 
Spintronics