Questions tagged [nonlinear-dynamics]
This tag is for questions relating to nonlinear-dynamics, the branch of mathematical physics that studies systems governed by equations more complex than the linear, $~aX+b~$ form.
251
questions with no upvoted or accepted answers
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views
Extension of Burgers' equation
I recently encountered a viscous Burgers' equation type PDE, but with the addition of a derivative-squared nonlinear term (in dimensionless form):
$u_t - u_{xx} + uu_x - u_x^2 = 0\,,$
where the ...
- 71
6
votes
0
answers
155
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Nature of ODE $\dot x=x^2-\frac{t^2}{1+t^2}$
Discuss the equation $\dot{x}=x^2-\frac{t^2}{1+t^2}$.
Make a numerical analysis.
Show that there is a unique solution which asymptotically approaches the line $x=1$.
Show that all solutions below ...
- 9,098
6
votes
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129
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Estimate on derivative of ODE solution with respect to parameters
Consider the ODE
$$
u'(t) = f(t,u,p), \qquad u(0) = v
$$
where $p$ is a control parameter, and let $u(t;v,p)$ denote the solution to the problem above for fixed $v$ and $p$.
It is apparently "...
5
votes
0
answers
444
views
What is correlation dimension, actually?
I'm taking a course on chaotic dynamical systems, and we're talking about attractors with non-integer correlation dimensions, but I can't seem to find a satisfactory definition for this concept. ...
- 1,016
5
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221
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What are the solutions for $y(t)\cdot\left(y'(t) + a\right)=-b\sin(t)$?
What are the solutions for $y(t)\cdot\left(y'(t) + a\right)=-b\sin(t)$?
It could be proben that there exists some solutions?
Are these solutions unique?
and obviously, which are these solutions? (...
- 1,534
5
votes
0
answers
289
views
Creating a SIIR (susceptible, infected, isolated, recovered) model using differential equations.
I wasn't too sure of where to post this since it's a mix of physics (dynamical systems), medicine, and mathematics but here it goes.
I am trying to model the current outbreak of Covid 19 using a more ...
- 73
4
votes
0
answers
170
views
Dynamics of a sliding cube on the $XY$ and $YZ$ planes
A cube with side length $a$, is initially placed with one vertex at the origin, and its faces parallel to the coordinate planes ($XY, XZ, YZ$) and totally lying in the first octant. Then its rotated ...
- 24.4k
4
votes
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51
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Find steady state of AIDS epidemic model
AIDS epidemic in a homosexual population
The following diagram shows the AIDS epidemic in a homosexual population:
Then the model can be described by
$$
\begin{gathered}
d X / d t=B-\mu X-\lambda c X ...
- 875
3
votes
0
answers
84
views
Are two separate limit cycles in a dynamical system possible
In all the examples I've seen before with two limit cycles, the limit cycles are always concentric (there is an unstable point at center, a stable limit cycle on the middle and an unstable limit cycle ...
- 77
3
votes
0
answers
157
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Convergence to locally stable equilibria
I am working with a system whose trajectories converge to the set of equilibria. I can characterize all the equilibria in a nice way and easily compute their stability and whether they are isolated or ...
- 268
3
votes
0
answers
44
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Lyapunov Exponent of Nonlinear Schrödinger Operator
I am interested in ideas for how to compute the maximal Lyapunov exponent of the nonlinear Schrödinger-Newton system given by
$$
\partial_t s(t, \vec{x}) = L(s(t,\vec{x}))
$$
where
$$
L(s(t, \vec{x})) ...
- 197
3
votes
0
answers
125
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Find the root of $x(t;A)=0$ for ordinary differential equation $\ddot{x}(t;A)=A(p(t)\dot{x}(t;A)+q(t)x(t;A))$
It is given an ordinary differential equation $\ddot{x}(t;A)=A(p(t)\dot{x}(t;A)+q(t)x(t;A))$ with real parameter $A$, initial conditions $x(t=0;A)=1,\dot{x}(t=0;A)=0$ for all real $A$. If $r(A)$ is ...
- 185
3
votes
0
answers
31
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Why are continuous partial derivatives up to order two (rather than one) of nonlinear autonomous (2D) systems sufficient for linear approximation?
In Boyce and Diprima's ODE's and BVP's (10th edition page 522), it says that for the nonlinear autonomous system $$x^\prime = F(x,y)\qquad y^\prime = G(x,y) \qquad\qquad\qquad (10),$$
"The system ...
- 695
3
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231
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Does heteroclinic intersections of the stable and unstable manifolds of different fixed points imply chaotic behavior?
Suppose we have a saddle fixed point of a map, then we have the stable and unstable manifolds of the saddle points. If the stable and unstable manifolds intersect (If they intersect once they will be ...
- 5,032
3
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133
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Lyapunov Stability for a Nonlinear, Time-varying system
I am currently trying to learn how to determine the stability of a solution using Lyapunov's Method for non-autonomous systems.
Say we are given a nonlinear system:
$$\dot{x_1}(t)=-x_1(t) + x_2(t)[...
