# Stability Definition and 46 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. ### What is the smallest value of angular displacement of the raft?

What is meant by "the smallest value of angular displacement of the raft from its original position during one cycle"? I understand that I am supposed to solve this problem using torques of the crane and and of the boxes, but I am totally confused by that "smallest angular displacement". If it...
2. ### I Error tolerant normal mode frequency

If a Hookean spring-mass system is made from one mass and a spring, to produce a system with a particular oscillation frequency, it's not a problem to use the propagation of errors concept to find how this frequency responds to small errors in the mass and spring constant. If a chain of...
3. ### Engineering Equilibrium and stability

When I use Lagrange to get the equations of motion, in order to find the equilibrium conditions I set the parameters q as constants thus the derivatives to be zero and then calculate the q's that satisfy the equations of motion obtained. In ordert to check about stability I think I need to add...
4. ### Equilibrium points doubt (ODE system solution)

Hi. I'm not sure about something related to the equilibrium points (or fixed points) of a non linear ode system solution. As far as I know, to check if an equilibrium point exists, I need to put the function of my ode system equal to zero. Then once the point is found, I can use it to evaluate...
5. ### Stability of a Driven Pendulum

I understand that when $$A_0 \gg g$$, the g term in the equation of motion can be dropped. The equation of motion then becomes $$\frac{d^2\theta}{dt^2}=-\frac{a_d(t)}{L}\sin\theta$$ But how can I show that the pendulum is stable for such case? I am totally clueless.
6. ### Comp Sci Finding the stability of a system using Python code

I'm given this code to plot the system for task 1 where my teacher have used some random numbers just to show us how the code can be used: The code makes a plot that is given below: So my professor just gave us this code and plot without saying anything more. All the students in the class...
7. ### Stability and concavity of the entropy function

I am struggling to understand Callen's explanation for stability, I understand that the concavity of S(U) must be negative because otherwise we can show that this means that the temperature increases as the internal energy decreases (dT/dU<0) but I cannot understand equation (8.1) which...
8. ### Capped pressure vessel - strength and stability

Hi, in mechanics of materials books one may easily find fomulas for stress and strain in thin- and thick-walled cylindrical pressure vessels subjected to internal pressure. However, it is assumed that they are open. So what are the formulas for stress and strain in capped vessels (with flat or...
9. ### I Stability of circular orbits in an arbitrary central force field

In this chapter, the stability of an object orbiting in a circular orbit of radius r_c in an arbitrary force field f is considered. The author arrives at the equation of a harmonic oscillator, for small deviations x from the circular orbit: \ddot{x} + \left[-3\frac{f(r_c)}{r_c} -...
10. ### Engineering Stability of the system

See the picture I am stuck at 12(1+2k)=0 So k=-1/2 for stability k must have value greater the - 1/2 which means there will no sign changes in rooth array and equation represents a stable system
11. ### Is this a good alternative definition of electron affinity?

traditional definition of electron affinity: the amount of energy released by an element in its gas form when gaining an electron second definition?: the stability gained by an element in its gas form when gaining an electron (e.g. halogens are more stable after gaining an electron, and when...
12. ### Euler Approximation Failure

I had thought it would be failure of structural stability since in structural stability qualitative behavior of the trajectories is unaffected by small perturbations, and here, even tiny deviations using ##h## values resulted in huge effects. However, apparently that's not the case, and I'm not...
13. ### A Coorbital question

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...
14. ### Electrical Engineering: Is this signal stable?

Homework Statement y(t) = x(1-5t) Homework Equations none The 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)...
15. ### I Why is the black hole stability conjecture so important?

I am working on a presentation for a course in general relativity and my topic is the stability of black holes. In many of the references and articles that I have found, the author asserts the importance of the conjecture but offers no reason. So I ask: Why is the black hole stability conjecture...
16. ### Why is a beam supported by two cables more stable than one?

Imagine a steel I-girder lifted by a single wire rope. If the rope is not perfectly centered along the girder length, the girder will rotate about a horizontal latitudinal axis through its centroid. Imagine a girder pair lifted by two wire ropes, with one rope on each side of the girder...
17. ### Calculating Miller Stability

<< Mentor Note:Thread moved from the Homework forums because it isn't really about homework >> Preface With my limited understanding of physics, I am unaware of what section of the forums this question would fall into. I am not doing this for school(though possibly it might help with that in...
18. ### How do I calculate the forces on a guitar stand (Tripod type)

I want to build a bass guitar stand for my guitar using wood, I'm modelling it after the image shown. The alterations I'm making are to increase the height of the legs , to place an attachable weight below the stand . This will allow the stand to have a lower center of mass and will greatly...
19. ### Equilibrium stability

Homework Statement The equation x^2+3x-0.45=0 has two solutions x1=0.14 and x2=-3.14, these are supposed to be the locations at which the electric charge is in equilibrium with two other charges, should I use the first derivative to see at which location the charge will be stable? Homework...
20. ### Determine the stability of a fixed point of oscillations

Homework Statement I have a system of coupled differential equations representing chemical reactions and given certain initial conditions for the equations I can observe oscillation behaviour when I solved the equations numerically using Euler's Method. However, then it asks to investigate the...
21. ### How to derive the slip angle for a trailer-car system?

I just watched this video that perfectly explains how to derive the equations for the slip angle of a car: However, how could this be done for a three axle system (car-trailer)? Would you have to express distance a (in the video) in terms of b (distance from centre of gravity to the second...
22. ### Finding all the forces in this system

Homework Statement [/B] I am trying to identify every force in this system and prove that it is in mechanical equilibrium when tan(θ)=⅓. Initially I had to solve it by finding when the derivative of potential with respect to θ was 0, but now I am just trying to resolve the forces. Homework...
23. ### In which temperature range does WF_6 melt?

Hi Everyone I'm studying material Engineering and I'm currently preparing chemistry for the summer exams. Now, there is an old exam question which I don't know how to solve: "In which temperature range does ##[W^{+VI}F_{6}^{-I}]## melt?" My solution: Well, the 18-Electron rule is not...
24. ### I The Sun's habitable zone and orbital resonaces/stability

Let's say there's five Mars/Earth massed planets orbiting a star like the sun between 0.6 AU and 2 AU, what orbital resonance configuration can they be into ensure maximum stability? Would adding gas giants to the system enhance stability?
25. ### A What keeps Rhea stable? Simulation shows Titan-assist escape [solved]

Hi everyone, I have created a simulation of the major bodies in the Solar System, using the exact positions, velocities, and masses etc. at midnight on Jan 10 (as provided by the NASA HORIZONS project). Using Newtonian gravity I numerically simulate the forces between all the bodies (with a...
26. ### LDO voltage regulator stability

I haven't done any Bode plots before, and so I'm reading about LDO stability and I came across this file, which I am trying to understand: http://www.ti.com/lit/an/slyt194/slyt194.pdf Under the stability analysis, it names the poles and zeroes. How are these calculated? I assume that the...
27. ### Droop control (load sharing in AC Mains distribution networks)

Hi, I have quite a superficial understanding of Droop control: http://www.openelectrical.org/wiki/index.php?title=Droop_Control https://en.wikipedia.org/wiki/Droop_speed_control I'm not after a super technically detailed explanation, but if anyone has any more detailed sources for how it is...
28. ### Engineering Which mechanical engineering jobs are the most stable?

From what I understand, there are some industries in Mechanical Engineering that are typically boom/bust (like oil and gas and aerospace). I wonder which ones are the most stable? I think HVAC is a possibility, but I don't know if it's the kind of thing that I would like to work on for my entire...
29. ### Combined Motion Gyroscope

I want to build a Self Balancing autonomous bicycle for which I need to know about CMG System,Can Any one please guide me to the basics of Gyroscope and CMG- Ebooks would be a great starts.
30. P

### B Why having a low center of mass guarantee stability?

I know that, i studied it, but i can't actually make a picture in my mind, i can't really figure out why, i mean the forces and the torques are the same no? What's the big deal? What am i missing?
31. ### A Possibility of a stable mesonic nucleus

Hello everyone, If a nucleus consisting solely of mesons has enough mesons in it, will it be stable? Mesons are bosons, therefore (unlike baryons) they aren't effected by the pauli exclusion principle, so they all can acquire the lowest energy state. In theory, if there are enough mesons, the...
32. ### Difference between Lyapunov and linear stability criteria

Dear all, Consider the connection of two electrical circuits. Both circuits, Z1 and Z2, are stable and only one of them is non-passive. I.e., the eigenvalues are located in the LHP but Re{Z2(jw)}<0 in a frequency range. For studying the closed-loop stability, you represent the linear system by...
33. ### Can the universe become stable with a new electroweak vacuum

The top and Higgs mass determination arose the old discussion about electroweak vacuum metastablity. There is an interesting fact that with available data the universe places in the edge of stable and meta-stable zone tends to be inside the meta-stable region. This conclusion confirms up to...
34. ### Find the limit cycle for this dynamical system

Homework Statement Consider the dynamical system: $$\dot{r}=-ar^4+ar^3+r^6-r^5+r^2-r~;~~\dot{\theta}=1$$ Find all fixed points and limit cycles for: a) ##~~a=2## b)##~~a<2## c)##~~2<a<2\sqrt{2}## Homework Equations Not applicable. The Attempt at a Solution For all three values/ranges...
35. ### How (un)stable are the Lagrangian points 1, 2 and 3?

A couple questions, please: I know that the Lagrangian points 1, 2 and 3 are unstable and special Lissajous orbits plus some station-keeping are required to place a spacecraft around them. But I was wondering if they are so totally unstable that they can't temporarily "capture" a passing...
36. ### Relation between energy conservation and numerical stability

Hi, Consider the conservation laws for an isothermal linear incompressible flow governed by the mass and momentum equation. The kinetic energy equation is then solved to see if energy conserved. Can anyone tell me if once it is shown energy is conserved, it implies that convergence is obtained...
37. ### Bowl Balanced on a Bowl

Homework Statement A solid hemisphere of radius b has its ﬂat surface glued to a horizontal table. A second solid hemisphere of different radius a rests on top of the ﬁrst one so that the curved surfaces are in contact. The surfaces of the hemispheres are rough (meaning that no slipping occurs...
38. ### Physics Challenge Question: Stability on a Cylinder

Homework Statement A solid cube of side ##l = r*pi/2## and of uniform density is placed on the highest point of a cylinder of radius ##r## as shown in the attached figure. If the cylinder is sufficiently rough that no sliding occurs, calculate the full range of the angle through which the block...
39. ### Advection equation stability for explicit scheme

Hello, I have a typical 1D advection problem where a cold fluid flows over a flat plate. I did an energy balance to include conduction, convection and friction loss and I got the PDE's for the fluid and the solid. I used finite differences to solve the system as T(x, t) for both fluid and...
40. ### Can a car's reaction to gas pedal be said to be stable ?

As per control theory, if a bounded input produces a bounded output then a system can be said to be stable. So assuming that I press my cars gas pedal such that it reaches a fixed position, then the reaction of my car would be to reach a corresponding velocity, and assuming the road to be even...
41. ### Stability of a circular orbit

Homework Statement In a classical model of a multi-electron atom, electrons are assumed to move in a modified electrostatic potential $V(r)$, given by; $$V(r)=\dfrac{-k}{r}e^{-r/a}$$ Show that the effective potential is ; $$V_e(r)=\dfrac{J^2}{2mr^2}+\dfrac{-k}{r}e^{-r/a}$$ Then show that...
42. ### Pendulum - Stability and fixed points

Homework Statement (a): Show the lagrangian derivative in phase space (b)i: Show how the phase space evolves over time and how they converge (b)ii: Find the fixed points and stability and sketch phase diagram (c)i: Find fixed points and stability (c)ii: Show stable limit cycles exist for T>ga...