# Instability

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In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds. [1] Not all systems that are not stable are unstable; systems can also be marginally stable or exhibit limit cycle behavior.

A system is a group of interacting or interrelated entities that form a unified whole. A system is delineated by its spatial and temporal boundaries, surrounded and influenced by its environment, described by its structure and purpose and expressed in its functioning.

In quantum physics, a bound state is a special quantum state of a particle subject to a potential such that the particle has a tendency to remain localised in one or more regions of space. The potential may be external or it may be the result of the presence of another particle; in the latter case, one can equivalently define a bound state as a state representing two or more particles whose interaction energy exceeds the total energy of each separate particle. One consequence is that, given a potential vanishing at infinity, negative-energy states must be bound. In general, the energy spectrum of the set of bound states is discrete, unlike free particles, which have a continuous spectrum.

In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance.

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In structural engineering, a structure can become unstable when excessive 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.

Structural engineering is a sub-discipline of civil engineering in which structural engineers are trained to design the 'bones and muscles' that create the form and shape of man made structures. Structural engineers need to understand and calculate the stability, strength and rigidity of built structures for buildings and nonbuilding structures. The structural designs are integrated with those of other designers such as architects and building services engineer and often supervise the construction of projects by contractors on site. They can also be involved in the design of machinery, medical equipment, and vehicles where structural integrity affects functioning and safety. See glossary of structural engineering.

In engineering, deflection is the degree to which a structural element is displaced under a load. It may refer to an angle or a distance.

In science, buckling is an instability that leads to structural failure. The failure modes can in simple cases be found by simple mathematically solutions. For complex structures the failure modes are found by numerical tools.

Atmospheric instability is a major component of all weather systems on Earth.

Atmospheric instability is a condition where the Earth's atmosphere is generally considered to be unstable and as a result the weather is subjected to a high degree of variability through distance and time. Atmospheric stability is a measure of the atmosphere's tendency to discourage or deter vertical motion, and vertical motion is directly correlated to different types of weather systems and their severity. In unstable conditions, a lifted thing, such as a parcel of air will be warmer than the surrounding air at altitude. Because it is warmer, it is less dense and is prone to further ascent.

## Instability in control systems

In the theory of dynamical systems, a state variable in a system is said to be unstable if it evolves without bounds. A system itself is said to be unstable if at least one of its state variables is unstable.

A state variable is one of the set of variables that are used to describe the mathematical "state" of a dynamical system. Intuitively, the state of a system describes enough about the system to determine its future behaviour in the absence of any external forces affecting the system. Models that consist of coupled first-order differential equations are said to be in state-variable form.

In continuous time control theory, a system is unstable if any of the roots of its characteristic equation has real part greater than zero (or if zero is a repeated root). This is equivalent to any of the eigenvalues of the state matrix having either real part greater than zero, or, for the eigenvalues on the imaginary axis, the algebraic multiplicity being larger than the geometric multiplicity.[ clarification needed ] The equivalent condition in discrete time is that at least one of the eigenvalues is greater than 1 in absolute value, or that two or more eigenvalues are equal and of unit absolute value.

Control theory in control systems engineering is a subfield of mathematics that deals with the control of continuously operating dynamical systems in engineered processes and machines. The objective is to develop a control model for controlling such systems using a control action in an optimum manner without delay or overshoot and ensuring control stability.

In mathematics, the characteristic equation is an algebraic equation of degree n upon which depends the solution of a given nth-order differential equation or difference equation. The characteristic equation can only be formed when the differential or difference equation is linear and homogeneous, and has constant coefficients. Such a differential equation, with y as the dependent variable and an, an − 1, ..., a1, a0 as constants,

## Instability in solid mechanics

Elastic instability is a form of instability occurring in elastic systems, such as buckling of beams and plates subject to large compressive loads.

Drucker stability refers to a set of mathematical criteria that restrict the possible nonlinear stress-strain relations that can be satisfied by a solid material. The postulates are named after Daniel C. Drucker. A material that does not satisfy these criteria is often found to be unstable in the sense that application of a load to a material point can lead to arbitrary deformations at that material point unless an additional length or time scale is specified in the constitutive relations.

## Fluid instabilities

Fluid instabilities occur in liquids, gases and plasmas, and are often characterized by the shape that form; they are studied in fluid dynamics and magnetohydrodynamics. Fluid instabilities include:

## Plasma instabilities

Plasma instabilities can be divided into two general groups (1) hydrodynamic instabilities (2) kinetic instabilities. Plasma instabilities are also categorised into different modes – see this paragraph in plasma stability.

## Instabilities of stellar systems

Galaxies and star clusters can be unstable, if small perturbations in the gravitational potential cause changes in the density that reinforce the original perturbation. Such instabilities usually require that the motions of stars be highly correlated, so that the perturbation is not "smeared out" by random motions. After the instability has run its course, the system is typically "hotter" (the motions are more random) or rounder than before. Instabilities in stellar systems include:

## Joint instabilities

The most common residual disability after any sprain in the body is instability. Mechanical instability includes insufficient stabilizing structures and mobility that exceed the physiological limits. Functional instability involves recurrent sprains or a feeling of giving way of the injured joint. [5] Injuries cause proprioceptive deficits and impaired postural control in the joint. Individuals with muscular weakness, occult instability, and decreased postural control are more susceptible to injury than those with better postural control. Instability leads to an increase in postural sway, the measurement of the time and distance a subject spends away from an ideal center of pressure. The measurement of a subject’s postural sway can be calculated through testing center of pressure (CoP), which is defined as the vertical projection of center of mass on the ground. Investigators have theorized that if injuries to joints cause deafferentation, the interruption of sensory nerve fibers, and functional instability, then a subject’s postural sway should be altered. [6] Joint stability can be enhanced by the use of an external support system, like a brace, to alter body mechanics. The mechanical support provided by a brace provides cutaneous afferent feedback in maintaining postural control and increasing stability.

## Notes

1. "Definition of INSTABILITY". www.merriam-webster.com. Retrieved 23 April 2018.
2. "Definition of BAROCLINIC INSTABILITY". www.merriam-webster.com. Retrieved 23 April 2018.
3. Shengtai Li, Hui Li. "Parallel AMR Code for Compressible MHD or HD Equations". Los Alamos National Laboratory.
4. Merritt, D.; Sellwood, J. (1994), "Bending Instabilities of Stellar Systems", The Astrophysical Journal, 425: 551–567, Bibcode:1994ApJ...425..551M, doi:10.1086/174005
5. Guskiewicz, K. M.; Perrin, David H. (1996). "Effect of Orthotics on Postural Sway Following Inversion Ankle Sprain". Journal of Orthopedic and Sports Physical Therapy. 23 (5): 326–331. doi:10.2519/jospt.1996.23.5.326. PMID   8728531.
6. Pintsaar, A.; Brynhildsen, J.; Tropp, H. (1996). "Postural Corrections after Standardised Perturbations of Single Limb Stance: Effect of Training and Orthotic Devices in Patients with Ankle Instability". British Journal of Sports Medicine . 30 (2): 151–155. doi:10.1136/bjsm.30.2.151. PMC  . PMID   8799602.