What is Stability: Definition and 522 Discussions

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.
In dynamical systems, an orbit is called Lyapunov stable if the forward orbit of any point is in a small enough neighborhood or it stays in a small (but perhaps, larger) neighborhood. Various criteria have been developed to prove stability or instability of an orbit. Under favorable circumstances, the question may be reduced to a well-studied problem involving eigenvalues of matrices. A more general method involves Lyapunov functions. In practice, any one of a number of different stability criteria are applied.

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  1. mollwollfumble

    A Are Retrograde Satellites the Key to Stable and Optimal Orbits?

    I was working on a proposal for a spacecraft , and suddenly realized that the ideal orbit may be a high inclination type of near-Earth coorbital called a "retrograde satellite" or RS orbit. Do you know of: * A person who can compute 100 years of coorbital stability using three body (sun...
  2. jchan79

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  3. A

    B Radioactive decay, Stability and Halflife

    As it's written in the following article, nuclear binding energy is always a positive number; thus it takes energy to disassemble a nucleus into its nucleons. ...The binding energy is always a positive number, as we need to spend energy in moving these nucleons, attracted to each other by the...
  4. Z

    How to calculate the stability and strength of a simple chassis?

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  5. Z

    Negative Charge Stability: Oxygen vs. Carbon/Nitrogen

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  6. Leonardo Machado

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  7. Boltzman Oscillation

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    Homework Statement y(t) = x(1-5t) Homework Equations noneThe Attempt at a Solution well I've never looked at the stability of a signal which has a time scale and shift. My guess is that it is stable as anything I can provide as input will output a bounded signal. Ex: if x(t) is u(t) then...
  8. dRic2

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  9. M

    Stability - calculations taken for granted in books

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  10. Vishesh Jain

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  11. G

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  12. S

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  13. F

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  14. bbbl67

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  15. D

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  16. Krushnaraj Pandya

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  17. Krushnaraj Pandya

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  18. A

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  19. T

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  20. A

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  21. R

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  22. C

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  23. mjtsquared

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  24. manu101

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  25. T

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  27. H

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  28. Brilli

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  29. Xeno777

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  30. J

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  31. K

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  32. Mary1995

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  33. peadar2211

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  34. MathematicalPhysicist

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  35. A

    Roll Stability of static structures

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  36. G

    B Concept of the stability of the speed of light

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  37. M

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  38. T

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  39. L

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  40. Alexander350

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  41. H Psi equal E Psi

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  42. T

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  43. Solomei

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  45. P

    I The Sun's habitable zone and orbital resonaces/stability

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  46. Z

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  47. G

    Magnetism - stability of a ring with a sphere inside it?

    Hey, is it possible to have a cylindrical ring (made of permanent magnet) and a magnetic sphere sitting inside it and staying in one place without touching the inner sides of the ring and not falling through? The object doesn't necessarily have to be a sphere. If it would work with any other...
  48. M

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  49. J

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  50. D

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