Mathematics Stack Exchange
Mathematics Stack Exchange

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.

  • Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.
  • In general, systems involving flows (heat, fluid, etc) demonstrate nonlinear dynamics, but they also show up in classical mechanics (e.g. the three-body problem, the double-jointed pendulum).
  • The method that is most used in nonlinear dynamics is Runge-Kutta.
  • Nonlinear dynamical problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature.
  • As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization).

For more details see https://en.wikipedia.org/wiki/Nonlinear_system

501 questions
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What is the most general Carathéodory-type global existence theorem?

I am looking for a general theorem that guarantees the existence of a global solution for an ODE system in $\mathbb{R}^n$ $$ \begin{equation} \left\{ \begin{aligned} x'(t) &= f(t, x(t)), \qquad t \in [a,b] \\ x(a) &= x_0 \end{aligned}…
A. Pesare
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What is meant when mathematicians or engineers say we cannot solve nonlinear systems?

I was watching a video on "system identification" in control theory, in which the creator says that we don't have solutions to nonlinear systems. And I have heard this many times in many contexts, related to control problems or nonlinear odes, etc.…
krishnab
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Stability analysis of the dynamical system $\ddot{x}+b\dot{x}+K x-\|\dot{x}\| \frac{x-x_i}{\|x-x_i\|^2}=0$ .

Consider the dynamical system described as: $$\ddot{z}+b\dot{z}+ K z-\|\dot{z}\| \frac{z-z_i}{\|z-z_i\|^3}=0$$ where $z=[x \ \ y]^T$, $K$ is a positive definite matrix and $b \in \mathbb{R}$, I made some simulations and based on the numerical…
abc1455
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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 boundary conditions require the solution to vanish at…
JonasB
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Does Asymptotic Stability Imply the Existence of a Lyapunov Function for a Nonlinear System?

For a linear time-invariant system $\dot x = Ax,$ the inverse Lyapunov theorem asserts that if the origin is asymptotically stable, then a Lyapunov function in the form $V(x) = x^\top P x$ for some positive definite function $P.$ Is there a similar…
Paul Wintz
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7
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1 answer

Generalisation of Index of a curve to higher dimensions

Im studying Non Linear Dynamics and Chaos from Strogatz's textbook. In the sixth chapter, while talking about non linear flows in 2 dimensions he introduces the index of a curve in a vector field and shows some beautiful properties that the index…
Aritra Das
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6
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Are there general solutions to quadratic, 2D, continuous, time-invariant dynamical systems?

I am a bit new to dynamical systems and don't know my way around terminology, so have had a hard time answering this for myself. I know the basics of theory for 2D linear, time-invariant systems, i.e., $$ \dot{x}=a_1x+a_2y \\ \dot{y}=b_1x+b_2y $$ I…
dang
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An ODE confusion

I was thinking about a ODE problem recently when I was reading about dynamical system. In school we used to solve the ODE problem $\frac{dx}{dt}=\sqrt{1-x^2}, x=0, t=0$ as $x=\sin(t),$ which will have the graph Now in dynamical system we can see…
Ri-Li
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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 this solution approach the line $x=$ $-1 .$ Show…
Ri-Li
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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 "well known", that the analyticity of $f$ implies that $x$…
Daniele Avitabile
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How does this expression follow algebraically from the last one?

I was reading this paper: Global stability for an HIV/AIDS epidemic model with different latent stages and treatment Everything is understood apart from on page 7 of the pdf (page 1486 in the document). How does the author algebraically go from the…
user644376
5
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1 answer

Why is this approximate solution correct?

Consider the following differential equation $$ y''=-y + \alpha y |y|^2, $$ where $y=y(x)$ is complex in general and $\alpha$ is a real constant such that the second term is small compared to $y$ ($||^2$ is the absolute square). Numerically I find…
user655870
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$f(n) = \frac{n^2 + n + 4}{2}, g(f(n)) = f(g(n))$ such that $g(n)$ is an integer.

Let $n$ be a strict positive integer. Lets define an integer sequence $f(n)$ : $$f(n) = \frac{n^2 + n + 4}{2}$$ so $$f(1) = 3$$ $$f := {3,5,8,12,17,23,30,38,47,...}$$ $$f(17) = 155$$ etc Notice $$3+2=5$$ $$5+3=8$$ $$8+4=12$$ $$12+5=17$$ etc and…
mick
  • 17,886
5
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1 answer

How to prove this theorem for the number of components of a filled julia sets?

If one finite critical point of $f(z)$ escapes to infinity by iterating, then the filled-in Julia set of $f(z)$ consists of infinitely many components. How to prove this ? I must admit I heard this in the context of polynomials so maybe there are…
mick
  • 17,886
5
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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. Multiple sources, including Wikipedia, define…
Frank Seidl
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