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. D

    Stability in the family of baryons

    Of all the baryons, why are protons the only known stable combination of quarks? Why are neutrons more stable by more than billions of times longer than the rest of the unstable baryons? If it is just their combination of quantum numbers, then, why do those combinations work out to being stable?
  2. D

    Convergence and stability in multivariate fixed point iteration

    Hi, I'm new to posting questions on forums, so I apologise if the problem is poorly described. My problem is solving a simulation of the state of a mineral processing froth flotation plant. In the form x@i+1 = f(x@i), f represents the flotation plant. f is a computationally intensive solution...
  3. X

    Equilibrium and Stability of a set of equations

    Homework Statement Find the equilibrium and determine its stability for the model of Sec 1.4. Assume x and y are both strictly positive (nonzero). Homework Equations Differential equations of section 1.4 (1/x)(dx/dt) = a1 - b1y (1/y)(dy/dt) = -a2 + b2x The Attempt at a Solution I solved...
  4. C

    Cylohexanol stability of conformations

    There is an equilibrium between the conformer in which OH is axial and the conformer in which OH is equatorial. axial - equatorial in apolar solvents, the equilibrium is less displaced to the right than in polar solvents. Any ideas? I think it has to do with entropy but not sure how...
  5. E

    Von Nuemann Stability Analysis CN scheme

    Homework Statement Perform a von Nueman stability analysis for the Crank-Nicholson FTCS scheme for the following ADE: dc/dt + u dc/dx = D d2c/dx2 - kc where c is the concentration of interest, u is the advecting velocity and D is the dispersion coefficient. We used a general forward...
  6. I

    Building a JFET Transistor Cascade for Gain, Linearity & Stability

    I need help on how to cascade two or three Jfet transistor and the calculation of the values to have a good amount of gain. The cascade also should help in providing good linearity, have less distortion and should stable good stability.Thanks, good help will really be appreciated by me.
  7. C

    Stability of a moon of hot jupiter in red dwarf system

    I was trying to calculate possible orbit of a potential exomoon of a hot Jupiter within habitable zone of a red dwarf. Everything (heat absorbed, reasonably within Hill sphere, far from Roche limit) seamed reasonable. However, I encountered one problem - hot Jupiter should be tidally locked. It...
  8. S

    Thermal stability vs melting point

    An ionic compound is more thermally stable when the metal cation is more reactive. So its harder to decompose. But then what's the difference between melting and decomposition? Also, when an compound is harder to decompose, does it mean its melting point is higher? Because even though Na2O...
  9. J

    Stability of the atom in QFT

    If I may, I would like to give this question another try, especially if guys with some cabala in QFT can address it. Is it possible to show in the relativistic quantum theory, that the hydrogen atom is stable? (Electron will not fall onto the proton)? I explain. In the non-relativistic...
  10. S

    What is the relationship between stability and the derivative of a fixed point?

    Dear friends, I want to find the conditions of stability of a fixed point. consider the function "f" iterates to obtain fixed point "a": x_{n+1}= f(x_n) for this dynamic system, the fixed point "a" is stable if we have: |f ^{\prime}(a)| < 1 Currently I'm working on a bit different...
  11. C

    Conditions of Stability for second order system

    Homework Statement Consider the transfer function H(s)=\cfrac{1}{a_{2}s^{2}+a_{1}s+a_{0}} where real-valued coefficients a_{2},a_{1}, a_{0} are arbitrary except that a_{2} is nonzero. Verify that the system is stable iff the coefficients a_{2},a_{1}, a_{0} have the same sign. Homework...
  12. R

    Understanding System Stability: BIBO Stability Question Explained

    Hi, I am trying to find out if a system is bounded (stable) or not, and I don't understand the process, I basically have these two functions, y1[n]=n2*x[n] y2[n]=x4[n] According to my notes, y1[n] is not stable, as the output is proportional to n2 which is not bounded, I understand...
  13. fluidistic

    Stability of Critical Points in a System of DEs

    Homework Statement Find the stationary solution(s) of the following system of DE and determine its stability: x'=x-x^3-xy^2. y'=2y-y^5-x^4y.Homework Equations x'=0, y'=0. Expansion of x and y around the critical point: x=x_0+\alpha, y=y_0+\beta.The Attempt at a Solution I solved x'=y'=0, this...
  14. fluidistic

    Stability of Stationary Solutions in a System of Differential Equations

    Homework Statement Find the stationary solution(s) of the following system of DE and determine its stability: x'=x^2+y^2+1. y'=2xy.Homework Equations x'=\frac{dx}{dt}=0, y'=\frac{dy}{dt}=0.The Attempt at a Solution I tried to google "stationary solutions of a system of DE" but didn't find...
  15. B

    Why Does a Spinning Top Stabilize After Wobbling?

    So I get that conservation of angular momentum makes a spinning top stable. Same mechanism behind gyroscopes. When you first spin a top, there's a lot of wobble (precession), but it quickly dissipates. Why does this happen? The lack of wobbling must be a lower energy state if it is reached...
  16. N

    Regarding jk flip flop Q' stability.

    Hi, I am trying to understand and analyze JK flip flop's (using RS flop flop i.e. using NOR gates). I have written out the characteristic table and when I give Q = 1, J=1 and K = 0 I am trying to analyze the Q' (Q complement) by giving the intial state to 0. I find that Q settles to 1 (i.e...
  17. N

    Finding Eigenvectors & Stabilizing 0,0 in System Stability

    For the system \dot{x}=y2 \dot{y}=x2 Both the eigenvalues are zero. How do I find the eigenvectors so that I can sketch the phase portrait and how do I classify the stability of the fixed point (0,0)?
  18. T

    How Do Stability and Linearity Determine System Behavior?

    confused!..system stability and linearity 1)-Is y(n)=cos{x(n)} a stable system?? and is the condition s=Ʃ|h(k)|<∞ for stability valid only for LTI systems? actually my book solves the given problem using the above method..but according to me the given system is not LTI SINCE ZERO I/P...
  19. F

    What is the stability proof for a PI controller in a formation flying system?

    I have two satellites flying in formation geverned by the equations of motion including J2 and drag. Now, one of the satellites has more drag than the other so they become appart. I designed a controller (proportional integral) which meassures the along track position between them and gives...
  20. P

    Understanding Lorenz Equations & Stability: A Homework Guide

    Homework Statement http://imageshack.us/photo/my-images/535/newpicture3.jpg/ Homework Equations Honestly i used the equation in Strogatz 10.3.3 so I am not sure how to do it for others. The Attempt at a Solution http://imageshack.us/photo/my-images/406/worke.png/...
  21. A

    Stability of Unstable Particles at High Energies: Third Generation

    Are particles that are not stable (for example top, bottom quarks; W and Z) when produced in particle colliders possibly stable at very high energies? Are third generation particles stable at high energies? Thanks, Mark
  22. K

    Cause of stability in counter-rotating gyros

    http://en.wikipedia.org/wiki/Gyro_Monorail#Principles_of_operation It appears that counter-rotating gyros can indeed provide stability without precession. Is this simply because of the fact that when you accelerate the rotation of the axis of adjacent counter-rotating spinning wheels you are...
  23. P

    Atomic Physics - Radioactive Decay and Stability

    Hi, Explain in terms of the number of nucleons and the forces between them, why argon-36 is stable and argon-39 is radioactive. My first doubt regards the number of nucleons. If a nucleon is the collective number of neutrons and protons, if we take carbon 12 for example, does it have 6 or...
  24. B

    Find the equilibrium points and their stability in the system

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  25. T

    Steady-State Stability and the Second Derivative

    Hi. I have a question about steady-state stability and the second derivative test. I have been reading about it in a book on mathematical modeling, and the section concerns differential equations. I believe this forum is more appropriate than "General Math," but let me know if it is not...
  26. D

    Stability of Equilibrium solutions to ODE

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  27. U

    Understanding Stability of an Object: CG vs CB & Metacentre

    Hi, As shown in the attachment, the different situations that occur in an object with 1. the CG above the Centre of buoyancy (CB) and 2. CG below the CB are shown. A. It is said that the more stable state is when CG is below CB...however, I don't understand this point, since as seen in...
  28. L

    Electrical stability of an atom

    Homework Statement As shown in class the electric potential energy of a water molecule is -5.33 eV. Which molecule is more electrically stable, H2O or CO? Why? Homework Equations EPE=Kc*q1*q2/r12 The Attempt at a Solution The water molecule's EPE is -5.33 eV. The carbon...
  29. T

    Stability of a barge- Fluid Mechanics

    Homework Statement A large barge of overall width 4m to be used at sea has a uniform empty weight per meter length of M=6327.19kg/m (length is into the page). This empty weight is uniformly distributed along the length of the barge. The center of gravity of the empty barge is at position G...
  30. D

    Stability of a Large Barge: Determining Stability Using Weight Per Meter Length

    Homework Statement The question is regarding the stability of a large barge. Overall width=4m Uniform weight per meter length of M= 6327.19kg/m (This empty weight is uniformly distributed along the length of the barge. Center of gravity of the empty barge is at position G, 2m above the flat...
  31. F

    Stability of Bicycles: Why is it Easier Moving?

    I have this burning question_Why is it that we can stabalize bicycle so easily when its moving but not when its still?
  32. T

    First Order ODE Stability

    Homework Statement In the market for a certain good, the price p(t) adjusts continuously in the presence of excess supply or demand: \frac{dp}{dt} = F(D(p)-S(p)) where F(0) = 0, F'>0. Obtain the condition for stability of the equilibrium price p* in terms of the slopes D'(p*) and S'(p*), and...
  33. D

    Stability of fast neutron reactors with liquid metal coolant.

    Is my understanding correct that the short term stability of fast neutron liquid metal cooled reactors is based primarily on the thermal expansion of the core, while the Doppler coefficient is far less significant factor, as the Doppler coefficient primarily affects the low energy neutrons? (The...
  34. H

    Bicycle Stability: Moving vs. Resting

    why is moving bicycle more stable than rest one?
  35. M

    Is Power Balance a Valid Criterion for Oscillating System Stability?

    Lets consider this definition: If the average power supplied to the system (absorbed power) is less than the average power lost by the system (emitted power) then the system is stable (during the time in which the power was averaged). Is such a definition of stability valid?
  36. M

    Conceptual question - radioactive nucleus stability

    Homework Statement A ^{238}_{92} U nucleus can catch a neutron with small kinetical energy. Then we'll get a ^{239}_{92} U nucleus. This nucleus has too many neutrons compared to the number of protons to be stable. How can the nucleus achieve a better balance between neutrons and protons...
  37. S

    Exploring the Tent-Map: Fixed Points & Stability

    Homework Statement The "tent-map" is given by: xn+1 = g(xn) where g(x) = 2x if 0 <= x<= 1/2 and g(x) = 2-2x if 1/2 < x <= 1 a) Find the fixed points and their stability. Draw a cobweb plot of the tent map to demonstrate that your stability calculations are correct. b) Find a period-2 orbit...
  38. B

    Mass speed molecular stability

    If you increase mass with speed does the forces that hold molecules’ together increase in per portion ? If not do the molecules’ fall apart with speed ? If the forces increase in per portion Why does the energy released from the muluation of those forces decrease with speed.
  39. L

    Stability of a linear system

    Homework Statement Determine the stability of the following linear system y(n) = 0.5x(n) +100x(n-2) - 20x(n-10) Homework Equations x(n) = 0.5\delta(n) S=\sum^{\infty}_{k=0}\left| h(k)\right| The Attempt at a Solution Z \left\{ 0.5x(n) +100x(n-2) - 20x(n-10) \right\} Z...
  40. P

    Stability of Products: Nuclear binding energy vs Enthelpy

    I want to get my head around this... Why is that in nuclear fusion, the formed nucleus is more stable because its nuclear binding energy/nucleon is HIGHER than the sum of its reactants But in general chemical reactions products are more stable if their enthalpy is LOWER than the sum of...
  41. M

    Test Stability using Routh Stability Method

    Homework Statement For a control system that has G(s)H(s) = \frac{1}{s^{2}*(s+\alpha)}Homework Equations 1 + G(s)H(s) = 0 The Attempt at a Solution Exam question i messed up . I really need to know the answer.
  42. J

    Improving the temperature stability of a temperature controlled water bath

    Hello, I have a temperature controlled water bath that is capable of controlling the temperature of the water to 0.2 degrees C. However this isn't good enough for my application and I require a greater level of stability. I want to place a metal box filled with oil inside the temperature...
  43. S

    Stability of the gaussian under addition and scalar multiplication

    If i have the mgf of X and the mgf of Y where X~N(mx,vx) and Y~N(my,vy) and X and Y are independent , then if i want to show that aX +bY ~ N(amx+bmy , a^2vx+b^2vy) how would i do this - need to be able to do the convolutions way and the mgf's way, for the mgfs way is it just, mgf(ax+by) =...
  44. S

    Showing stability under scaling and additivity of distriubtions

    Homework Statement I need to show that if X ~ r(a1,B) Y ~ r(a2,b) where r means gamma distribution then if X and Y are independent i) X+Y ~ r(a1+a2,B) ii) cX ~ r(a1,cB) Homework Equations The Attempt at a Solution i) i use the mgfs of x and y and ended up with mgf(x+y) =...
  45. L

    Island of Stability: Answers to Questions

    Okay, so even though I really don't need to, I do a LOT of research on astronomy, physics, math, etc. I was doing research on the extended periodic table, and super heavy unstable elements. I was quickly turned to the Island of Stability, and I get the theory but have a few questions. 1...
  46. R

    Crank Nicolson Stability

    Hello, I was wondering about teh stability of the Crank Nicolson method as compared to the forward time center space central difference method of solving differential equations. In particular, I was wondering the the stability criteria used for the central difference method, stability =...
  47. R

    Stability Triangle of Forklift

    Hi I came across this: http://www.free-training.com/osha/forklift/Physics/54.htm Here I don't understand why the forklift will remain stable only when the CG is inside the triangle. If the CG is outside the triangle but still within the square formed by the four wheels of the forklift why...
  48. C

    How to find the stability of equibrium points

    Homework Statement I want to know if i got answer to C correct http://img822.imageshack.us/img822/7804/equilbrium.gif Homework Equations The Attempt at a Solution [PLAIN][PLAIN]http://img703.imageshack.us/img703/508/equilbriumsol.gif
  49. M

    Can Nuclide Stability Be Determined by Atomic Weight or Radioactivity?

    I have the impression that when discussing stability of nuclides that there are two different usages of the term. 1. Atomic weight divided by nucleon count: The lower the number, the more stable is the nuclide. Iron ending up being the most stable. 2. Radioactivity: All nuclides that...
  50. B

    What has bandgap gotta do with binding energy and stability?

    Hi, Could someone clarify the following statement please? "ZnO has a bandgap of 3.4 eV at room temperature and a free exciton binding energy of 60meV which is much larger than the room temperature thermal excitation energy (25meV) making them stable at rtp." Does it mean that at rtp, we...
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