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.
25
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Increasing convergence rate using Optimal Control and Pontryagin Maximum Principle
My question is in addition to Tuning the optimal control synthesized according to the Pontryagin maximum/minimum principle and choosing the cost function, but requires help from the mathematical side ...
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4
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2
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Solving the recurrence relation $y_{n+1}=y_n+a+\frac{b}{y_n}$
I am hoping to obtain a closed-form solution or an asymptotic result for the recurrence relation
$$y_{n+1}=y_n+a+\frac{b}{y_n}$$
Any help would be very much appreciated!
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It is possible for a scalar finite-duration continuous system to achieve an infinite speed (in finite-time)? How if true? Why not if false?
It is possible for a scalar finite-duration continuous system to achieve an infinite speed (in finite-time)? How if it true? Why not if it false? (Please read first the restrictions of the system I am ...
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2
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1
answer
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Adaptive step size for Euler Method - How to create?
I think Euler's Method is a great way to simulate ODE:s. It's not the most accurate, but it's the fastest and simplest.
Euler's Method is usually used with fixed step size, where $k$ is the step size ...
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2
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Understanding Uniqueness of solutions of differential equations - nonlinear ODEs - pendulum example
Understanding Uniqueness of solutions of differential equations - nonlinear ODEs - pendulum example
I am trying to understand If the nonlinear ODE of the classical equation for the pendulum with ...
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6
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1
answer
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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)...
user644376
5
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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? (...
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1
answer
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Intuition behind dense orbits
I'm having some difficulties discerning the difference between attracting sets an attractors in my nonlinear systems course. The definition we've been given is that attractors are attracting sets that ...
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3
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1
answer
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Does this dynamical system have another conserved quantity?
For the 3D system of ODEs:
$$\begin{eqnarray}\dot{x} &=& -\beta x y \\
\dot{y} &=& \beta x y + \hat{\beta} z y - \delta y \\
\dot{z} &=& -\hat{\beta} z y + \delta y,
\end{...
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3
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Example of Topologically Mixing map on $k$-dimensional cube
Let $k,M$ be positive integers. Is there a simply explicit example of a topologically mixing map on:
The "cube" $[-1,1]^k$?
The "disc" $\{x \in \mathbb{R}^k: \|x\|\leq 1\}$?
And what are the ...
user683848
1
vote
1
answer
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Problem with the continuous equivalent of Newton's method optimization
In the arcticle Fixed-Time Stable Gradient Flows: Applications to Continuous-Time Optimization I found an interesting formula and its properties. The screenshot of the page from the article I was led ...
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1
vote
1
answer
100
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Expansion of nonlinear functions with damping properties in exponential series
I am working on solving nonlinear differential equations and found such a solution with exponential properties.
$\frac{dx}{dt}=\frac{d}{dx}(sech(x)^2)$
The solution of which is:
$x(t) = \sinh ^{-1}\...
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1
vote
1
answer
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Unsolvable characteristic system ODE as a part of PDE solution?
I'm trying to solve the following PDE:
$$F(x_1,x_2,u,p_1,p_2)=\text{ln}(x_2)p_1+x_2up_2-u=0 \ \ \ \ \ \ p_i=\partial_iu(x_1,x_2)$$
Where the initial conditions are:
$$\begin{cases}x_1(t)=t+1 \\x_2(t)=...
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vote
0
answers
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Which way of solving from nonlinear control to choose?
I have a nonlinear system:
\begin{cases} x'=f(x)+u \\ y=f(x) \end{cases}
where $f(x)$ - gradient of some one-extremal function (for example $f=e^{-(x)^2}$), i.e. $\frac{df}{dx}$.
Task:
I want ...
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1
vote
1
answer
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Solving $x'=-\text{sgn}(x)\sqrt{|x|}$: Uniqueness of solutions of finite duration
Show that $$x(t) = \frac{\text{sgn}(x(0))}{4}\left(2\sqrt{|x(0)|}-t\right)^2\cdot\theta\!\left(2\sqrt{|x(0)|}-t\right)$$ is a solution to$$x'=-\text{sgn}(x)\sqrt{|x|}.$$
Is the solution unique?
This ...
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