In numerous fields of study, the component of instability within a system

System

System is a set of interacting or interdependent components forming an integrated whole....

 is generally characterized by some of the outputs or internal states growing without bounds. Not all systems that are not stable

Stability theory

In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions...

 are unstable; systems can also be marginally stable

Marginal stability

In the theory of dynamical systems, and control theory, a continuous linear time-invariant system is marginally stable if and only if the real part of every eigenvalue in the system's transfer-function is non-positive, and all eigenvalues with zero real value are simple roots...

 or exhibit limit cycle behavior.

In linear control theory

Control theory

Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...

, a system is unstable if any of the roots of its characteristic equation

Characteristic equation

Characteristic equation may refer to:* Characteristic equation , used to solve linear differential equations* Characteristic equation, a characteristic polynomial equation in linear algebra used to find eigenvalues...

 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

State space (controls)

In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations...

 having either real part greater than zero, or, for the eigenvalues on the imaginary axis, the algebraic multiplicity being larger than the geometric multiplicity.

In structural engineering

Structural engineering

Structural engineering is a field of engineering dealing with the analysis and design of structures that support or resist loads. Structural engineering is usually considered a specialty within civil engineering, but it can also be studied in its own right....

, a structure can become unstable when excessive load is applied. Beyond a certain threshold, structural deflection

Deflection

Deflection or deflexion may refer to:* Deflection , the displacement of a structural element under load* Deflection , a technique of shooting ahead of a moving target so that the target and projectile will collide...

s magnify stresses

Stress (physics)

In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

, which in turn increases deflections. This can take the form of buckling

Buckling

In science, buckling is a mathematical instability, leading to a failure mode.Theoretically, buckling is caused by a bifurcation in the solution to the equations of static equilibrium...

 or crippling. The general field of study is called structural stability

Structural stability

In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by C1-small perturbations....

.

Atmospheric instability

Atmospheric instability

Atmospheric instability is a condition where the atmosphere is generally considered to be unstable and as a result the weather is subject to a high degree of variability through distance and time...

 is a major component of all weather systems on Earth.

Fluid instabilities

Fluid

Fluid

In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....

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

Fluid dynamics

In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

 and magnetohydrodynamics

Magnetohydrodynamics

Magnetohydrodynamics is an academic discipline which studies the dynamics of electrically conducting fluids. Examples of such fluids include plasmas, liquid metals, and salt water or electrolytes...

. Fluid instabilities include:

• Ballooning mode instability (some analogy to the Rayleigh–Taylor instability); found in the magnetosphere
  Magnetosphere
  A magnetosphere is formed when a stream of charged particles, such as the solar wind, interacts with and is deflected by the intrinsic magnetic field of a planet or similar body. Earth is surrounded by a magnetosphere, as are the other planets with intrinsic magnetic fields: Mercury, Jupiter,...
  
  
• Atmospheric instability
  Atmospheric instability
  Atmospheric instability is a condition where the atmosphere is generally considered to be unstable and as a result the weather is subject to a high degree of variability through distance and time...
  
  
  • Hydrodynamic instability or dynamic instability (atmospheric dynamics)
    • Inertial instability; baroclinic instability; symmetric instability, conditional symmetric or convective symmetric instability; barotropic instability; Helmholtz or shearing instability; rotational instability
  • Hydrostatic instability or static instability/vertical instability (parcel instability), thermodynamic instability (atmospheric thermodynamics
    Atmospheric thermodynamics
    Atmospheric thermodynamics is the study of heat to work transformations in the earth’s atmospheric system in relation to weather or climate...
    
    )
    • Conditional or static instability, buoyant instability, latent instability, nonlocal static instability, conditional-symmetric instability; convective
      Convective instability
      In meteorology, convective instability or stability of an airmass refers to its ability to resist vertical motion. A stable atmosphere makes vertical movement difficult, and small vertical disturbances dampen out and disappear...
      
      , potential
      Convective instability
      In meteorology, convective instability or stability of an airmass refers to its ability to resist vertical motion. A stable atmosphere makes vertical movement difficult, and small vertical disturbances dampen out and disappear...
      
      , or thermal instability
      Convective instability
      In meteorology, convective instability or stability of an airmass refers to its ability to resist vertical motion. A stable atmosphere makes vertical movement difficult, and small vertical disturbances dampen out and disappear...
      
      , convective instability of the first and second kind; absolute or mechanical instability
• Bénard instability
• Drift mirror instability
• Kelvin–Helmholtz instability (similar, but different from the diocotron instability
  Diocotron instability
  A diocotron instability is a plasma instability created by two sheets of charge slipping past each other. Energy is dissipated in the form of two surface waves propagating in opposite directions, with one flowing over the other. This instability is the plasma analog of the Kelvin-Helmholtz...
  
   in plasmas)
• Rayleigh–Taylor instability
• Plateau-Rayleigh instability
  Plateau-Rayleigh instability
  The Plateau–Rayleigh instability, often just called the Rayleigh instability, explains why and how a falling stream of fluid breaks up into smaller packets with the same volume but less surface area. It is related to the Rayleigh–Taylor instability...
  
   (similar to the Rayleigh–Taylor instability)
• Richtmyer-Meshkov instability
  Richtmyer-Meshkov instability
  The Richtmyer–Meshkov instability occurs when an interface between fluids of differing density is impulsively accelerated, e.g. by the passage of a shock wave. The development of the instability begins with small amplitude perturbations which initially grow linearly with time...
  
   (similar to the Rayleigh–Taylor instability)

Plasma instabilities

Plasma

Plasma (physics)

In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...

 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 cluster

Star cluster

Star clusters or star clouds are groups of stars. Two types of star clusters can be distinguished: globular clusters are tight groups of hundreds of thousands of very old stars which are gravitationally bound, while open clusters, more loosely clustered groups of stars, generally contain less than...

s 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:

• Bar instability of rapidly-rotating disks
• Jeans instability
  Jeans instability
  In physics, the Jeans instability causes the collapse of interstellar gas clouds and subsequent star formation. It occurs when the internal gas pressure is not strong enough to prevent gravitational collapse of a region filled with matter...
  
  
• Firehose instability
  Firehose instability
  The firehose instability is a dynamical instability of thin or elongated galaxies. The instability causes the galaxy to buckle or bend in a direction perpendicular to its long axis. After the instability has run its course, the galaxy is less elongated than before...
  
  
• Gravothermal instability
• Radial-orbit instability
• Various instabilities in cold rotating disks

External links

• eFluids Fluid Flow Image Gallery