Articles of dynamical systems

Population Dynamics model

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 […]

Definition of Hamiltonian system through integral invariant

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 […]

Bifurcation Example Using Newton's Method

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?

Can linear systems always be represented as differential or difference equations?

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 […]

Lyapunov Stability of Non-autonomous Nonlinear Dynamical Systems

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 […]

Poincare-Bendixson Theorem

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?

Noncausal dynamical system

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 ($n = 0, m = 1$): […]

What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?

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 […]

How can a Markov chain be written as a measure-preserving dynamic system

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 […]

Definition of strong mixing and definition of measure-preserving

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 […]