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.

View More On Wikipedia.org
Recent contents Top users
  1. A

    How can electrodynamics prove the stability of electron

    in quantumdynamics, the stability of electron of H atom is a hypothesis. but if electrodynamics is right in quantum theories, electron circling a H atom will emit energy, then how can electroquantum theory give a sound prove of the stability of electron? thanks.
  2. S

    Help with stability of full valence shell

    Hello, I am wondering about the real explanation that atoms have a tendency to have higher stablity with a full valence shell. What forces are at work here besides the 'happy' atom. -Scott
  3. B

    Is the 'branching out' pattern always present in bifurcation diagrams?

    I have another question about stability, it kind of leads on from the other topic that I posted. On bifurcation diagrams, you typically see a series of 'forks'(or branches) and the number of forks for a given interval usually increases as you move from end to the other doesn't it? I'm...
  4. B

    Difference equations and stability

    Hi I am unsure about stability of fixed points here is an example. x_{n + 1} = x_n There are fixed points at x = 0 and x = 1. In general when talking about difference equations and whether a fixed point is stable or unstable, does this refer to points in a neighbourhood of those...
  5. happyg1

    Numerically Stable Formulas for x >= 0 and x < 0

    Hi, here's my question...We have to determine whether or not the formulas down there are numerically stable for the cases where x >=0 and x<0. I say that the formulas are stable for x>=0 but not for x<0 because you are subracting numbers that could be close to each other. My problem is that I...
  6. S

    News Is Kim Jong-Il's Behavior a Sign of Mental Instability?

    I've heard many people on these forums claiming Il is crazy, a psychopath, or other such things. I havn't seen or heard any such reports about him having any such condition anywhere. How true is this? How much is just fear mongering?
  7. V

    News Stability of Anarchy: Let's Continue Here, Smurf

    Let's continue here, Smurf.
  8. A

    Can a space elevator be stabilized without reaching geostationary orbit?

    Hi. This may be really basic, but would appreciate your help. Is a tower whose height reaches a geostationary orbit at 35,000km's altitude or so more stable that one which doens't quite reach that height? So, say you are building a tower, decide you'd like it more stable, so should you go...
  9. R

    Static margin and longitundiual stability

    hi i've a problem with the definition of static margin and the role of horizontal stablizer on static margin, is ther anyone to help me thanks erkan
  10. A

    Stability of transuranic elements

    I remember hearing somewhere that one of the reasons people are trying to make elements which high atomic numbers is to see if they can be made to have long half-lives, such as uranium. In effect, they're virtually stable. However, I don't see how that can be. Here's why. Suppose element...
  11. M

    Flagpole Stability: Calculating Maximum Tension

    A uniform, horizontal flagpole of length 5.00 m and with a weight of 195 N is hinged to a vertical wall at one end. A stuntwoman weighing 600 N hangs from its other end. The flagpole is supported by a guy wire running from its outer end to a point on the wall directly above the pole...
  12. B

    Thermal & Kinetic Stability: What's the Difference?

    Within the context of the equation: CH4(g) +2O2(g) -> CO2(g) + H2O(l) delta H = -890 KJ What is the difference between Thermal and Kinetic stability? I knew the answer before, but i forgot now. Forgive me for just memorising the answer, but its a temporary solution. Any help would be...
  13. J

    Stability of a given structure composed of three rods and a string

    Three rods with length l and mass m are arranged like the figure in the attached file. There connections are without friction. One rod is horizontal and the other two parallell and in the same plane. An elastic string connects point A and B. The tension in the string is proportional to its...
  14. N

    Can Mass Distribution and Spring Constants Keep a Hovering Platform Stable?

    Hi there. I have got very good help from this forum in the past and I have yet another question. Take a square platform with a thruster at each corner. Put a weight in the centre of it. Now, with the weight in the centre the platform is stable but as you move the weight to one side the...
  15. M

    Stability of Area of Triangle?

    Consider the following algorithm for computing the area of a triangle with sides of length A, B, and C: S = (A+B+C) / 2 Area1(A,B,C) = sqrt(S(S-C)(S-B)(S-A)). A) Explain why this approach is unstable, and illustrate with an example. B) A student proposes the following less...
  16. F

    Designing a Maglev Train: Feasibility and Stability

    I am doing a project for my physics course...I have to design a device that uses electromagnetism. I'm thinking of doing something "different" rather than the boring bells/radios etc.. Would a small-scale model of a maglev train be feasible in everyone's opinion? I am planning to build a...
  17. S

    Frequency Stability: What Does 2 Parts in 10E10 Mean?

    Does anyone know what it means to say that the frequency stability of a laser is 2 parts in 10E10, I gather that it means that the frequency shifts 0.02nm each side of the mean wavelength. But I can't find any resources that use the term parts.
  18. G

    Cell Membranes & Protein Stability

    Hi. What prevents proteins from falling out of cell membranes? This question has been bothering me for sometime... and I cannot find an answer to this. Please help :-p
  19. G

    Stability of spin top (tippy top)

    Stability of "spin top" (tippy top) I'm looking for a decent discussion of the stability of a spin top (sometimes called the tippy top), which shows that the stable configuration when spinning is the "inverted" one (spinning on the stick). Any help would be appreciated.
  20. B

    Is the Triangular Lamina Stable When Suspended from Point C?

    A uniform triangular lamina ABC where AB = 1m and BC = 2m and it is right angled at B. It is suspended from C and hangs freely under gravity. Calculate the angle between BC and the downwards vertical. Need help fast pls cheers:smile:
  21. Monique

    News EU Stability Pact: Why Germany & France Exempt?

    Why don't Germany and France have to comply to the rules? According the the stability pact, a country in the EU is not allowed to have a budget deficit larger than 3%. France has 3.5% and Germany 4%, in a meeting last night the other EU countries voted on the issue. Spain, Finland, Austria...
  22. J

    Protons: Stability, Decay & Half-Life

    Do protons decay or are they completely stable? If they do then what is their half-life?
Back
Top