India, November 11, 2017

The course 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. The
course will focus on different aspect of parametric
instabilities in magnetic systems, at zero-, one- and two-
spatial dimensions.

Description:

 

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