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

471 questions
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What is the model of Shilnikov Bifurcation

There is this model created by Shilnikov and described in the paper: https://www.researchgate.net/publication/220580167_Shilnikov_bifurcation The problem is that I do not understand how the system of equations (1) gives the homoclinic loop, which…
Axiom Feraas
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finding a closed orbit for an oscillator equation

Consider the oscillator equation $$\displaystyle\ddot{x}+F(x,\dot{x})\dot{x}+x=0$$ where $F(x,\dot{x})<0$ if $r\leq a$ and $F(x,\dot{x})>0$ if $r\geq b$ where $r^2=x^2+\dot{x}^2$. Show that there is at least one closed orbit in the region…
am_11235...
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Checking stability of a fixed point

If the fixed point is hyperbolic, then it is said that linearisation gives the correct result . Is there an intuitive way of understanding why this is so ? And for marginal cases, when the fixed points are found to be centres ot stars by…
user157588
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logistic map and stable cycle

Show that the logistic map $x_{n+1}=Ax_n(1-x_n)$ has stable $2$-cycle for all $A>3$. I am a newbie in non-linear dynamics and chaos theory. All I have been able to find out about the map is that it has equilibrium points at $x^*=0$ and…
am_11235...
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State space representiation for given diff. eq.

I am looking for the state space representation for the following differential equation: $m_{1}\ddot{x}+c_{1}\dot{x}+(k_1+k_{1,p}x^{2})x=F_{0} \ cos \ \omega t$ Rewriting this gives: $\ddot{x} =…
user463102
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Generalized alignement index of classic Lorenz system?

I am reading about generalized alignment index (GALIs) as chaos indicator. However, I have been looking around for a while now to see an example of this applied on to the classic Lorenz attractor, but to my surprise I couldn't find any! I would…
Axel Wang
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Dynamics - Normal Form

Im working through the problems of Strogtz`s Nonlinear Dynamics and Chaos. On problem 4.3.5, it does not ask, but is it possible to find a normal form for it? The equation is the following: $$ \dot{\theta} = \mu+\cos(\theta)-\cos(2\theta)$$ which…
Lyu
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Convergence in the Hegselmann-Krause model

In classical opinion dynamics, we define $N$ agents. Agent $i$ has an initial opinion sampled from a distribution with mean $\mu$. Let agent $i$'s initial opinion be $y_i(0)$ (a random variable of that distribution). Agents interact with one another…
B2VSi
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Proof of Unstability

Let assume that the dynamical system $x_{t+1}=f(x_t)$ (with $f(0)=0$) is globally unstable (which means from any initial condition $x\neq0$, the states go to infinity). Can we prove that $x_{t+1}=f(x_t)+y_t$ (where $y_t$ is norm bounded $||y_t||
Amir Khazraei
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Phase portrait of a nonlinear system

I have an Lotka Volterra type of system as seen below; $$\begin{align} x' &= 10 x y -1/2 x - 1/10 x^2 \\ y' &= y - y^2 -x y \end{align}$$ I would like to plot the phase portrait of the system. Critical points (points where $dx/dt$ and $dy/dt$ are…
vyaman
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How to convert the following second order ode to a system of first order odes?

I am given the following ode: $$x''=mg\sin(\theta)+k(L-\sqrt{x^2+h^2})\left(\frac{x}{\sqrt{x^2+h^2}}\right)-bx'$$ where $x$ is a function of $t.$ How do I convert this equation to a system of first order odes? I tried the usual technique of…
Student
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