Proposed Q&A site for windows 10 operating system users with questions and issues about use of the operating system.

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Proposed Q&A site for enthusiastic users of Valve's gaming platform, Steam

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Proposed Q&A site for developers , IT Pro's, Enterprise/Solution/Domain Architects, DevOps working on the Azure.

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Proposed Q&A site for developers who have purchased, or plan to purchase InvenSense chips and/or software to create their inventions. I would imagine typically small time developers and large companies.

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Given the system x' = ax$^c$ - $\phi$x, y' = by$^c$ - $\phi$y, $\phi$ = ax$^c$ + by$^c$ Part a was just showing that the derivative is 0, which I was able to just fine. The second part of the question says: Show that all trajectories starting in the positive quadrant are attracted to the invar...

The definition of the Lyapunov exponent is $\lambda = \lim_{n\rightarrow \infty}\left \{ \frac{1}{n} \sum_{i=0}^{n-1} ln\left | f'\left ( x_{i} \right ) \right |\right \}$. A point $x_{i}$ is superstable if the Lyapunov exponent tends to negative infinity when n tends to infinity. By constructi...

The dynamic of the system is showed as below: \begin{cases} \dot x=y \\ \dot y =2\delta y-x+1\ \end{cases} $$[x(t^+),y(t^+)]^T=[-x(t),0]^T\\ for \ x(t)<0 \ and \ y(t)=0, (0<\delta<1)$$ The exact piecewise solution of the above equation: $$x(t)=e^{\delta t}\{\{x(0)-1\}cos(\omega t)+\frac1\omega \{...

$$ \dot{x} = \sigma(y-x) \\ \dot{y} = r \ x - y - xz \\ \dot{z} = -\beta z + xy $$ For a Lorentzian system, the node at (0,0,0) is stable for value of parameter $r<1$. I found that it turns in to a saddle node when $r$ is more than 1, i.e. i found that for $r>1$, some eigenvalues are more tha...

I am having a system of differential equations as follows, now this is a non-linear system - $\dot{x}(t) = -0.28571(x + f_{1}) + 0.00057(g_{1} - z) $ $\dot{y}(t) = \frac{1}{c_{2}}(-2.4*y + 0.000101*f_{1}*y + 2.4*g_{2} -c_{2}*\dot{h_{1}} - 0.000101*h_{1}*x - 0.006*x)$ $\dot{z}(t) = -\dot{g_{1}}...

How can I prove that there exists a solution to dynamical system presented below? $$ \dfrac{dT}{dt} = \lambda - \alpha T + rT\bigg(1 - \dfrac{T+I}{T_{max}}\bigg) - kVT \\ \dfrac{dI}{dt} = kVT - \beta I \\ \dfrac{dV}{dt} = N \beta I - \gamma V \\ $$

Let $a,b \in \mathbb{R^2}$ and consider $U: \mathbb{R^2}-\{a,b\} \to \mathbb{R}$ be a smooth function which satisfies $\limsup_{|q| \to \infty} |U(q)| =1.$ Consider the system of ODEs for $(p(t),q(t)) \in \mathbb{R^2} \times (\mathbb{R^2} - \{a,b\}):$ $$ \begin{cases} \dot{p} = \nabla U(q(t)) & \...

I am having difficulties understanding the construction of Morse-Smale systems. They start with $M$ compact and connected smooth manifold, then they say there exists an inmersion (or embeddement) $i: M \to \mathbb{R}^n$. Then given $t \in \mathbb{R}$, lets say $t = 1$ for now. They define the...

This is probably a relatively straight-forward question but I haven't been to source for a simple enough to understand illustration of a trapping region and bounded trapping region. Definition: A trapping region of any dynamic system is a region such that every trajectory that begins in the...

I'm doing a project on a chaotic periodically driven system, and I would like to construct a Poincare map of the system. Everywhere says that for a periodically driven system, you simply choose you Poincare section by samplying the position and velocity of the system with a period equal to that o...

The dynamic of the system is showed as below: $$ \begin{cases} \dot x=y \\ \dot y =2\delta y-x+1\ \end{cases}$$ $$[x(t^+),y(t^+)]^T=[-x(t),0]^T\\ for \ x(t)<0 \ and \ y(t)=0, (0<\delta<1)$$ The typical chaotic attractor: The exact piecewise solution of the above equation: $$x(t)=e^{\delta t}\{...

H All, Is somebody here familiar with circle map-Arnold family? As i have to prove the 1 and 2, but there is no much info on internet to help understand. if someone knows please help! thanks

Is there an equivalent of KL divergence between probability measures that are not absolutely continuous with respect to the Lebesgue measure? I am interested in rigorous notions of typical and atypical (rare) trajectories in a dynamical system. We can assume uniform hyperbolicity for simplicity ...

Proposed Q&A site for IT Pros. From the help desk to the network infrastructure. There's much to discuss in the IT world. So why not discuss it here?

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Proposed Q&A site for esoteric language enthusiasts and esoteric language designers

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