Stability theory in the context of "Frequency response"

In mathematical systems theory, a multidimensional system or m-D system is a system in which not only one independent variable exists (like time), but there are several independent variables.

Important problems such as factorization and stability of m-D systems (m > 1) have recently attracted the interest of many researchers and practitioners. The reason is that the factorization and stability is not a straightforward extension of the factorization and stability of 1-D systems because, for example, the fundamental theorem of algebra does not exist in the ring of m-D (m > 1) polynomials.

The Mandelbrot set (/ˈmændəlbroʊt, -brɒt/) is a two-dimensional set that is defined in the complex plane as the complex numbers ${\displaystyle c}$ for which the function ${\displaystyle f_{c}(z)=z^{2}+c}$ does not diverge to infinity when iterated starting at ${\displaystyle z=0}$, i.e., for which the sequence ${\displaystyle f_{c}(0)}$, ${\displaystyle f_{c}(f_{c}(0))}$, etc., remains bounded in absolute value.

This set was first defined and drawn by Robert W. Brooks and Peter Matelski in 1978, as part of a study of Kleinian groups. Afterwards, in 1980, Benoit Mandelbrot obtained high-quality visualizations of the set while working at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York.

Control theory is a field of control engineering and applied mathematics that deals with the control of dynamical systems. The aim is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any delay, overshoot, or steady-state error and ensuring a level of control stability; often with the aim to achieve a degree of optimality.

To do this, a controller with the requisite corrective behavior is required. This controller monitors the controlled process variable (PV), and compares it with the reference or set point (SP). The difference between actual and desired value of the process variable, called the error signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects which are also studied are controllability and observability. Control theory is used in control system engineering to design automation that have revolutionized manufacturing, aircraft, communications and other industries, and created new fields such as robotics.

In mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation.

In dynamical systems instability means that some of the outputs or internal states increase with time, without bounds. Not all systems that are not stable are unstable; systems can also be marginally stable or exhibit limit cycle behavior.

In structural engineering, a structural beam or column can become unstable when excessive compressive load is applied. Beyond a certain threshold, structural deflections magnify stresses, which in turn increases deflections. This can take the form of buckling or crippling. The general field of study is called structural stability.

In a dynamical system, bistability means the system has two stable equilibrium states. A bistable structure can be resting in either of two states. An example of a mechanical device which is bistable is a light switch. The switch lever is designed to rest in the "on" or "off" position, but not between the two. Bistable behavior can occur in mechanical linkages, electronic circuits, nonlinear optical systems, chemical reactions, and physiological and biological systems.

In a conservative force field, bistability stems from the fact that the potential energy has two local minima, which are the stable equilibrium points. These rest states need not have equal potential energy. By mathematical arguments, a local maximum, an unstable equilibrium point, must lie between the two minima. At rest, a particle will be in one of the minimum equilibrium positions, because that corresponds to the state of lowest energy. The maximum can be visualized as a barrier between them.

In plasma physics, plasma stability concerns the stability properties of a plasma in equilibrium and its behavior under small perturbations. The stability of the system determines if the perturbations will grow, oscillate, or be damped out. It is an important consideration in topics such as nuclear fusion and astrophysical plasma.

In many cases, a plasma can be treated as a fluid and analyzed with the theory of magnetohydrodynamics (MHD). MHD stability is necessary for stable operation of magnetic confinement fusion devices and places certain operational limits. The beta limit, for example, sets the maximum achievable plasma beta in tokamaks.