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.
As we noted above, stability is all about the solution to the homogeneous equation.
For the equation
$$y''+by'+ay=0\tag{3}$$
we have discriminant
$$\Delta = b^2-4a\tag{4}$$
and the roots are
$$r=\frac{-b\pm\sqrt{b^2-4a}}{2}\tag{5}$$
We have three cases.
Case 1 (Distinct Real Non-Complex...
Hello,
I'm interessed by the following LTI system with control u :
x' = Ax +Bu and x(k+1) = Ax(k) + Bu(k).
In this paper (section 3): https://proceedings.neurips.cc/paper/2020/file/9cd78264cf2cd821ba651485c111a29a-Paper.pdf
They seems to say that only A needs to be stable to get a...
I have 2 bulks of emulsion (lotion, cream, yoghurt, sauce): 1kg and 0.030g. While in transit (plane and truck) the bigger bulk separated (oil pooled on top) and the smaller one stayed as is, with no changes to the appearance of the product.
In general, larger volumes of emulsion are more prone...
I am developing a Taekwondo body protector scoring system using a textile piezoelectric sensor. The sensor is designed to detect mechanical force and convert it into an electrical charge. However, it is sensitive to temperature variations, leading to potential inaccuracies and instability in the...
Dear experts,
I´m searching for some method to determine whether any 2D or 3D truss is stable without solving complex matrix equations. I want to implement such method in a simple computer program to discard any a priori non-stable trusses for further analysis.
Do you know any book or...
Although Ethyl chloride has more alpha hydrogens, cation formed by Benzyl chloride would show resonance and therefore show higher stability and faster rate, shouldn't that be more reactive?
Folks,
I'm trying to formulate the stability problem in the incompressible inviscid limit and find the dispersion relation in the Couette flow regime. As shown, the 2 infinite plates move one against the other, unlike the "standard" case where one plate is static and the second moves. I'm trying...
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...
Have these articles been discussed here previously? I could not find it but my search skills suck.
Kerr stability for small angular momentum
https://arxiv.org/abs/2104.11857 (just 800 pages)
Recently (31th May this year)
Wave equations estimates and the nonlinear stability of slowly rotating...
Hello there,
My attempted explanation/solution is :
“Torsional strain is caused by the tendency of the electron clouds in the interacting groups to repel each other, making it relatively difficult to rotate a group towards and through another group, as the repulsive force provides resistance to...
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...
I have a question understanding the reasoning in the book.
The book says in one dimension F=-dU/dr(p.185). From this, the system is stable at distance a when U'(a)=0 and U''(a)>0 where U is differentiated with respect to r.(p.217)
My question arises from the instance of a pendulum where a...
Iron (Fe-56) is in terms of nuclear energy spent, which seems equivalent to saying its nuclides are the most tightly-bound. Does this also make Fe-56 the most stable nucleus, and is nuclear potential energy to stability a general correlation? Do more-stable nuclei generally have less nuclear...
DLVO theory gives the curve of potential energy vs distance of two colloid particles. Potential energy curve is derived for colloids being only electrostatically stabilized and not sterically.
Looking at the image below which shows potential energy curve, we can see two local minima and one...
Unlike emulsions, microemulsions are thermodynamically stable. Its stability is often explained by entropy changes brought about by dispersing liquid in another liquid, however this can't be the whole story behind its stability since dispersing liquids also happens in regular emulsions and they...
Hi,
(This question is part of the same example as a previous post of mine, but I have a question about a different part of it)
I was looking at a question from an exam for a course I am self-teaching. There is a sub-question which asks us to find the values of a parameter for which the 2-cycle...
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...
Suppose I have an object consisting of a hemisphere of radius r and a cone of radius r and height h. The shapes are glued to each other on their faces and the object is set standing on its hemisphere side. Depending on the value of h, the center of gravity for the system will change.
I have...
Before I say something I want to kindly ask any possible participants to refrain from attacking the personal symptom explanation as untrustworthy or made up as some have done before in another thread. Although I will express my personal side in the thread as such, I am looking for answers from...
Hi,
Sorry in advance if the vocab is incorrect, I'm translating this from french.
In the correction, it's written that the 4th carbocation is the less stable due to the attracting inductif effect of the carbonyle group.
But isn't this carbocation stabilised by the mesomer effect of the C=O bond...
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...
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.
Hi,
I have to determine which molecule ( 2 and 3) is more stable.
I tried to determine the stability using the inductive effect, but when I tried to that, I ended with molecule 2 being more stable than molecule 3 since:
x3 ch3 will push electrons towards c+
Whilst in the 2nd molecule the is a H...
This is going to be a rather simple question about the understanding of covalent bonds.
Let's take the simplest molecule - H2 which is gas.
According to quantum mechanics, the two atoms in the molecule each share 1 electron with the other atom and those 2 electrons exist in the form of...
At constant pressure Enthalpy change is equal to heat exchange and we say that "if Enthalpy Change is negative then product formed is stable",
I am not able to make sense of this statement as change in Enthalpy tells us only about heat exchange but internal energy is function of both Work and...
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...
We know that for a non-rigid body, the most stable type of rotation of it is the rotation about the axis with the maximum momentum of inertia and thus the lowest kinetic energy. However, for this question involving a rigid body, the most stable axis is the one with the lowest moment of inertia...
Homework Statement:: time step must be greater than stability criterion
Relevant Equations:: stability criterion= h^2/4 x alpha
Hello. I have had to do 2 MATLAB codes based on the 2D Heat diffusion equation using the Explicit Finite Method. In those codes, the time step must be greater or...
If we just prove (https://www.physicsforums.com/threads/does-every-object-rotate-around-its-center-of-gravity.998359/) that object don't rotate about CG ,why then center of gravity must be ahead of center of pressure ,for yaw stability?
Can you explain physics behind this phenomen?
Hi,
I have a question that I am quite confused about. Please note this is at the undergraduate level.
Question: Given the transfer function with inverse multiplicative uncertainty \bar G (s) = \frac{G(s)}{1+\Delta \cdot W(s) \cdot G(s)}
and the fact that the system is connected in feedback...
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...
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...
Hi,
I was just working through a closed loop stability problem in Control Theory and I don't really understand how the answer has arrived at the solution so quickly.
Problem: We are working with the closed loop transfer function:
$$ P(s) = \frac{K(s) G(s)}{1+K(s)G(s)} $$, where $$ G(s) =...
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} -...
Consider the classical Heisenberg model without an external field which is defined by the Hamiltonian:
\mathcal{H} = -\sum_{ij} J_{ij} \vec{s_i}\vec{s_j}
where J_{ij} > 0 describes the coupling between the spins \vec{s}_i \in \mathbb{R}^3 on some lattice. (Is there a way to use tex...
Now that is an evil question! The more interesting case looks like when the cylinder is upright, so that is what I focused on. I defined two coordinates; ##\theta##, the angular displacement of the cylinder from the vertical and ##d##, the distance between the point on the cylindrical axis which...
I have a square column to be filled with gravel to avoid being moved or tipped over by wind. The column is about 1 ft x 1 ft x 8ft tall. It is sheathed with a covering and hollow inside. What is the optimum fill height to avoid the wind tipping over an unattached single column. And the...
Here is an image of the structure
I know that cyclopropyl methyl carbocation is exceptionally stable due to an effect so called dancing resonance which takes place because of lot of strain in cyclopropyl ring and vacant p orbital of Carbon attached with the ring.
So I think this is a similar...
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
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...
Given the following Controller equation Gol(s) and Plant equation Dol(s) for an open loop system the transfer function can be expressed as a ratio of polynomials where:
Gol(s) = b(s)/a(s)and Dol = c(s)/d(s).
For the open loop system the transfer function Tol = Gol(s)Dol(s) = b(s)c(s)/a(s)d(s)...
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...
I guess the resonator will be stable when both focal lengths of lens and curved mirror meet. The curved mirror is fixed and it's focal length:
$$f_{curved} = \frac{R_{2}}{2} = 50 \, mm.$$ Then the d should be: $$d=f_{lens}+f_{curved} = 100 \, mm.$$ I think that's also the distance for which the...
Hello all
I was hoping someone could help shed some light on understanding an equation for floating bodies.
I am trying to work out the distance between B and M shown in the sketch below:-
I have been given the equation:-
BM = I/V
BM = is the distance from center of buoyancy to the meta...
Hi
I need help from PF scholars to figure out one difficulty in understanding stability of nucleus. Nuclei remain unaffected during chemical reactions taking place even at very high temperatures and pressures. But their binding energy figures are not that high. Such chemical reactions have heat...
I am trying to understand attracting, Liapunov stable, asymptotically stable for given coupled system. I don't have any Liapunov function. Just two coupled systems such as
##\dot{x} = y##, ##\dot{y} = -4x##
or sometimes normal systems
##\dot{x} = -x##, ##\dot{y} = -5y##
How can I approach...
Not sure if this off-topic ,but I think it would fit in Stem Career guidance.
I'm currently a 2nd year physics major in Thailand and have been losing sleep thinking about my future in academia for days.
Yeah just like many of you guys here,
I was inspired to pursue physics by many of great...