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. W

    Stability against small perturbation.

    Hello, I am reading the book, The Quantum Theory of Fields II by Weinberg. In page 426 of this book (about soliton, domain wall stuffs), we have Eq(23.1.5) as the solution that minimizes Eq(23.1.3). The paragraph below Eq(23.1.5), the author said "The advantage of the derivation based on...
  2. O

    MHB Determining Stability of Equilibrium Point for x'=y-x3-xy2, y'=-x-x2y-y3

    i have this system x'=y-x3-xy2 y'=-x-x2y-y3 i worked it out and found the equilibrium point to be 0. how do i determine whether it is stable, assymp stable or not stable
  3. M

    Benzene Stability: Bond Dissociation vs. Aromaticity

    Might be a stupid question but got to ask. The bonds of benzene have a bond dissociation energy in between that of a single C-C bond and an alkene yet its stability is much higher that expected due to aromaticity/e- delocalization. Is stability not necessarily reflected in bond dissociation...
  4. A

    Understanding Quadcopter Stability: Exploring Uniquely Unstable Technology

    Hello everyone, Recently I've been looking at quadcopter technology. While looking through literature, I noticed that most people mention that a quadcopter is inherently unstable but no reasoning is provided. I looked a bit at the EOMs but that's a big mess that I don't have time for yet. I...
  5. C

    Stability Condition for Circular Orbit

    Homework Statement Show that the stability condition for a circular orbit of radius a, i.e. f(a) + \frac{a}{3} (\frac{df}{dr})_{r=a} < 0 is equivalent to the condition \frac{d^2V(r)}{dr^2} > 0 for r=a where V(r) is the effective potential given by V(r) = U(r) +...
  6. K

    MHB Interval of stability, Lyapunov exponent

    For the function fc(x)= (6/x) + (x/2) -c, generate an estimate of the Lyapunov exponent for at least one c value chosen from each of the following intervals : (note 0 <= c <= 3) a) the interval of stability of the fixed point b) the interval of stability of the 2-cycle c) the interval of...
  7. K

    MHB Fixed point, interval of existence, & stability

    Investing function fc(x) = (6/x)+(x/2)-c where 0<= c <=3 a) Use alegbra to find the positive fixed point (in terms of c) and identify its exact interval of existence b) Use algebra and calculus to find the exact interval of stability of the fixed point c) Use algebra to find the points of the 2...
  8. D

    Question about system stability

    If a system transfer function has a pole equal with 1 , j, cos(pi/4)+j*sin(pi/4) in other words its location is on the margin of the unit circle, it`s the system stable? In my opinion it`s not stable because we have a sum of 1 which doesn't converge but I am not sure. Everywhere I've read it...
  9. G

    Stability theory (Boundary Layer)

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  10. W

    A predictor-corrector method and stability

    A predictor-corrector method for the approximate solution of y'=f(t,y) uses \begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P \end{equation} as predictor and \begin{equation} y_{n+1}-y_{n}=\frac{h}{2}(f_{n+1}-f_{n}) \tag C \end{equation} IF (P) and (C) are used in PECE mode on the...
  11. R

    MHB A predictor-corrector method and stability

    A predictor-corrector method for the approximate solution of $y'=f(t,y)$ uses \begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P \end{equation} as predictor and \begin{equation} y_{n+1}-y_{n}=\frac{h}{2}(f_{n+1}-f_{n}) \tag C \end{equation} IF $(P)$ and $(C)$ are used in PECE mode on the...
  12. W

    Implicit Euler scheme and stability

    Find the fixed points of the implicit Euler scheme \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1}) \end{equation} when applied to the differential equation y'=y(1-y) and investigate their stability? => implicit Euler scheme \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1})...
  13. R

    MHB Implicit Euler Scheme and stability

    Find the fixed points of the implicit Euler scheme \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1}) \end{equation} when applied to the differential equation $y'=y(1-y)$ and investigate their stability? => implicit Euler scheme \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1}) \end{equation}...
  14. R

    MHB Absolute stability of finite difference scheme

    The Finite difference scheme: \begin{equation} y_{n+3}-y_{n+1}= \frac {h}{3}(f_{n}-2f_{n+1}+7f_{n+2}) \end{equation} Deduce that the scheme is convergent and find its interval of absolute stability(if any) => the first characteristic polynomial is then \begin{equation} ρ(r)= r^3 -r...
  15. R

    MHB Runge-Kutte: stability of fixed points

    Show that the explicit Runge-Kutta scheme \begin{equation} \frac {y_{n+1} -y_{n}}{h}= \frac{1}{2} [f(t,y_{n} + f(t+h, y_{n}+hk_{1})] \end{equation} where $k_{1} = f(t,y_{n})$applied to the equation $y'= y(1-y)$ has two spurious fixed points if $h>2$.Briefy describe how you would investigate...
  16. B

    Gibbs Free Energy A Measure of Stability

    Hello everyone, I am having a little difficulty understand precisely what Gibbs free energy is. I have read in my textbook that a negative change in Gibbs free energy implies that the substance under consideration will react/change spontaneously. As such, the more negative the Gibbs free...
  17. J

    Analysing bode plots. Stability

    Hi all, I would be greatful is someone could kindly enlighten me as to the correct interpretation of the appended bode plots. My understanding when interpreting bode plots is that we desire 0 gain where the phase is equal to or exceeds 180 degrees (marginally stable / unstable). In...
  18. A

    Finding Pole-zero pattern of transfer fcn and Stability of LTI system

    Homework Statement The transfer function of an LTI system H(s) = (s^2 + 2)/(s^3+2s^2+2s+1) Find the followings i) pole-zero pattern of H(s) ii) Stability of the system iii) Impulse response h(t) Homework Equations Zero for which H(s) = 0 & Pole is for which H(s) = ∞ The...
  19. S

    Cyclohexene radical isomers stability

    The enthalpies of creating a cyclohexene radical isomers are: ORTHO: 444 kJ/mol META: 361 kJ/mol PARA: 401 kJ/mol The meta-isomer is most stable (that is the reason for the formation of meta product in radical addition). Para/meta isomers energies seem obvious - both carbons...
  20. X

    Control Systems - stability and settling time

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  21. N

    Linear System Stability Analysis: Matrix Blocks, Integral Block Included

    Homework Statement I'm having to figure out if a system is asymptotically stable, stable, or unstable. I am given the system block diagram. However, each constant block is actually a matrix. Also, there is an integral block thrown in there... Homework Equations The Attempt at a Solution In a...
  22. L

    Stability of the Classical Rutherford Atom: A Hydrogen Example

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  23. S

    Pressure regulator model: how to study its stability

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  24. B

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  25. K

    Stability of Closed Time-Like Curves: Current Status?

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  26. H

    Quark Stability: Light vs Heavy - Why Study?

    I'm a new comer to study the hadron physics.Why light quarks are more stable than heavy quarks and which one is easy to study? why? Thank you
  27. P

    Differental equation system and Lyapunov stability

    Homework Statement Example: x'=y-x^3 y'=-x-y^3 Homework Equations The Attempt at a Solution Linear system x'=y y'=-x Is stable because Det(P-\lambdaE)=\lambda2+1 \lambda1,2=+-i So if I am not mistaken,than Ishould use Lyapunov stability,because the linear system is stable and I can't say...
  28. E

    Liquid Slosh and Vehicle Stability

    Hi, I'm working on an invention that eliminates liquid slosh in partially filled liquid containers and allows for center of gravity control. I'm wondering if there is a market for such an apparatus where elimination of all slosh is needed. I am aware of baffles, sponges, and diaphragms. A...
  29. N

    Interstellar gas cloud stability

    Disclaimer: I'm not a physicist I've never quite grasped interstellar gas clouds (i.e. the material for new stars) and how they work. If they were too sparse, then you'd expect them to just dissipate. If they were too dense, then you'd expect them to collapse spontaneously. But yet they seem...
  30. M

    Proving Lyapunov Stability for \(\dot{x} = Ax + B(t)x\)

    A question I am doing hints that the solution (y,\dot{y}) = (0,0) of \ddot{y} - \frac{2}{t}\dot{y} + y = 0 is unstable. I believe (although I am not 100% sure) that is true however I am struggling to prove it. I can rewrite the equation as a system of equations in matrix form to get \dot{x} =...
  31. P

    Eigenvalues and stability - question.

    Hi, Homework Statement How may I determine whether a system is stable if its input is equal to its output, hence yielding a system(transfer) function equal to 1? Furthermore, could an eigenvalue zero characterize a stable system? I am attaching three examples where I am asked to determine...
  32. M

    How Does CCD Offset Affect Laser Beam Exit Angle?

    hi, I'm measuring my laser pointing stability with a focusing lens and a CCD. Could you guys tell me what would the relationship be between the offset from focal point in CCD and the angle with which the beam exits the laser? CCD is in focus.
  33. A

    Lack of stability after radiation

    U = Th + alpha rays 92p 90p 146n 144n I have heard that if n/p ratio exceeds 1.56, the substance becomes radioactive. Now, for Uranium, n/p ratio is 1.59. It gives out alpha rays in order to gain stability and in turn forms Thorium. n/p...
  34. WaaWaa Waa

    How stability is achieved in a bicycle?

    I have always been under the impression that I totally understood the mechanics working behind the stability of a bicycle i.e. i) the gyroscopic effect of the spinning wheel ii) weight of cycle and rider and iii) the centrifugal force acting on the CG when the bicycle follows a curve path. If...
  35. D

    MHB Plotting runge kutta 4 stability region

    How can I plot the runge kutta 4 stability region? I know on the i axis the max is \(\pm 2\sqrt{2}\). The plot makes a heart type shape. I don't know how to plot it though but would like to.
  36. M

    Delocalisation stability - primarily an entropy or enthelpy issue?

    This question was prompted by reflecting after reading the standard textbook explanation that "the greater acidity of RCOOH vs ROH is due to the greater stability of the delocalised RCOO- ion causing the position of equilibrium to be further to the right". The equilibria can be written as...
  37. A

    How Can Numerical Stability Be Achieved in Unsteady Laminar Flow Equations?

    I took a CFD class last semester (had to leave school though due to personal garbage). I am making a come back this fall and as some extra credit I am trying to numerically solve the unsteady laminar flow equation in a pipe. The equation is \dot{U} + U'' + K = 0 where dots denote the time...
  38. P

    What is the Island of Stability in Nuclear Physics?

    I've heard this expression in nuclear physics: the "island of stability." I know it has to do with the stability of a heavy transuranium atom (at least i think so), but what precisely does that expression mean? And what does it have to do with quantum mechanics? Why is this "island" there...
  39. P

    Job stability and money in experimental physics and for postdocs

    I'm in my last year of college as an undergraduate physics B.S. and have so far tried astronomy and materials science research internships and haven't liked either of them that much. I enjoy the theory behind astrophysics but don't enjoy programming all day. On the other hand, I like the...
  40. hilbert2

    Can small changes in fundamental constants affect the properties of water?

    Suppose we have a matrix A that has eigenvalues λ1, λ2, λ3,... Matrix B is a matrix that has "very small" matrix elements. Then we could expect that the eigenvalues of sum matrix A + B would be very close to the eigenvalues λi. But this is not the case. The eigenvalues of a matrix are not...
  41. marcus

    Stability of Minkowski and deSitter Spaces in GR

    Minkowski space and deSitter space have been shown to be stable in GR under small perturbations. Perturbations do not intensify in higher frequency modes--these solutions don't go haywire and develop black holes all over the place. Piotr Bizon has shown that Anti-deSitter (AdS) space is not...
  42. Z

    MHB Stability, phase portrait, bifurcations

    I am stuck with another one -- Assume that f(x) has the following graph: (for graph please see the attachment) Consider the (1-dimensional) ODE: X’ = f(x): (a) Find all the xed points, and study their stability. (b) Draw the phase portrait of the system, as well as the graphs of the...
  43. P

    Why does regularity bring stability?

    Hi PF, I've been wondering why lattice structures form in metals and in salts. Why do fcc or bcc structures reduce the energy of a system so that regular lattices are favorable over those whose atoms are randomly placed? Thanks, Pillow
  44. fluidistic

    Stability, Helmholtz free energy mathematical relation proof

    Stability, Helmholtz free energy mathematical relation "proof" Homework Statement I must show that \left ( \frac{\partial ^2 F}{\partial V ^2} \right ) _T=\frac{\frac{\partial ^2 U}{\partial S^2} \frac{\partial ^2 U}{\partial V ^2} - \left ( \frac{\partial ^2 U}{\partial S \partial V} \right...
  45. K

    Is Lead Iodide Precipitate Stable Over Time?

    About 6 years ago, I made a precipitate of Lead Iodide in water by mixing Potassium Iodide and Lead Nitrate. I have kept the precipitate in a test tube. Would anybody know how stable is the precipitate over time. The temperature has been mostly between 20 and 32 degrees depending on the...
  46. tom.stoer

    Stability of atoms in QM / QED

    It is often stated that quantum mechanics is able to explain the stability of atoms. I think most explanations are cheating b/c the compare apples and oranges. There are two reasons in classical theory which indicate that atoms should be unstable: A) there is no minimum for the orbit; the...
  47. H

    Need help determining feasibility of designing a stability system

    Hello everyone! This is my first post here, and I'm a recent graduate (within 1yr) of a BS in Mechanical Engineering. I have always been a car nut, and I'm interested in designing, or installing and tuning, a stability control system for my car. I recently purchased a 2012 Subaru...
  48. Kudaros

    MATLAB Droplet Profile in Matlab- ODE stability

    Hello, I'm currently modeling the profile of a droplet (sessile drop, axisymmetric) in matlab. I've coded differential equations, applied the solver, and I get a reasonable result, except that it spirals continuously. The ODE's in question are: \frac{dx}{ds}=cos(\theta)...
  49. E

    MATLAB Stability issues of ODE solutions using Matlab

    Hello, guys I am struggling with attaining stability values for u in solving the diffusion equation. The stability of u depends on the value of r from : D=1000; r0=1000; std=1.0; tau=1.0; IP=2500; %initial pressure % % Radial grid and inhomogeneous term nr=51; dr=r0/(nr-1)...
  50. C

    Stability of Orbits Homework: Analyzing Central Force Motion

    Homework Statement A particle of mass m moves under a central force ##\mathbf{F}(\mathbf{r}) = -\frac{\mu}{r^2} e^{-kr} \hat{r}##. The particle undergoes motion in a circle if ##h^2 = (a\mu/m)e^{-ka}##. I have shown that if ##u(\theta) = 1/r,## then the orbit eqn for ##u(\theta)## becomes...
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