I am currently searching for population dynamics models. Concerning animal population growth, I have found the following so far : Growth models for fish Predator-prey : Lotka-Voltera and Nicholson-Bayeux models Several species in competition for a resource Do you know any other mathematical models ? My aim is to simulate the dynamics of a human […]

I’ve read that Poincare’s integral invariance can be used as a definition of a Hamiltonian system. That is to say, if $g^t$ is a phase flow satisfying $$\oint_{\gamma} \omega = \oint_{g^t \gamma} \omega$$ with $\omega = p_{1}dq_{1}+\ldots+p_{N}dq_{N}$ for any closed path $\gamma$, then there exists a function $H = H(q,p)$ such that $\dot{q} =\frac{\partial H}{\partial […]

I am studying dynamical systems as part of a research project. I have been using Newton’s Method and studying the dynamic properties. Does anyone know where I could find a relatively simple example of bifurcation using Newton’s Method?

On my note, it was written that linear systems can always be represented as either differential equations or difference equations. I forgot the source of the quote. But I am not sure if it is correct. For example, for a linear time-invariant system, its output is the convolution of the input and the system’s impulse […]

Let $\mathbf{F}:X\times\mathbb{R}^{+}\to X$ be a non-autonomous dynamical system, which is governed by $\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x}, t, u)$, viz, \begin{equation} \begin{split} \dot{x}_1 &= x_2 – 314.2 \\ \dot{x}_2 &= 122\sin{x_1}\cos{x_1} – 154.8x_3\sin{x_1} – 0.5x_2 + 201.2 \\ \dot{x}_3 &= 0.64\cos{x_1} – 0.8x_3 + 0.2u \end{split} \end{equation} where $\mathbf{x} = \mathbf{x}(t) = [x_1(t), x_2(t), x_3(t)]^T \in X […]

Can someone sketch some ideas of how to use the Poincaré-Bendixson Theorem to prove that there must be a fixed point contained inside a periodic orbit?

The differential equation $$a_ny(t)^{(n)} + \dots + a_0y(t)^{(0)} = b_mu(t)^{(m)} + \dots + b_0u(t)^{(0)} $$ with $a_i,b_i \in \mathbb{R}$ and $y,u:\mathbb{R}\to\mathbb{R}$ describes a time-independent, linear, SISO system. Why is this system noncausal (that means not physically realizable) if $m > n$? For instance, this equation describes a noncausal system ($n = 0, m = 1$): […]

I’m trying to solve the system $$ \begin{matrix} & \frac{df_1}{dt} = kf_1+lf_2 \\ & \vdots \\ & \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = kf_{N-1}-lf_N \end{matrix} \tag{1}$$ with $f_1 (0) = f_0$ and $\forall n \neq 1 \ [f_n(0)=0]$, where $k$, $l$ and $f_0$ are real positive constants. The system may also […]

From http://masi.cscs.lsa.umich.edu/~crshalizi/notabene/ergodic-theory.html irreducible Markov chains with finite state spaces are ergodic processes, since they have a unique invariant distribution over the states. (In the Markov chain case, each of the ergodic components corresponds to an irreducible sub-space.) By “ergodic processes”, I understand it to be the same as “ergodic measure-preserving dynamic system”, if I am […]

I have a few questions regarding measure-preserving dynamical systems $(X,\mathcal{A},\mu,T)$. 1) The definition of measure preserving is always stated as $$\mu(T^{-1}B)=\mu(B),$$ for all $B$. I believe this neither implies nor is implied by the similar statement $$\mu(TB)=\mu(B).$$ Is this correct? The problem lies in the fact that $TT^{-1}B \subseteq B \subseteq T^{-1}TB$ may be strict […]

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