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        Questions tagged [ds.dynamical-systems]

        Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.

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        Characterization of Time-homogeneous flows for conditional expectation

        Let $X_t,Y_t$ be $\mathbb{R}^d$-valued processes. It is well known that for every $t\geq 0$, and every bounded function $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$, there exists a Borel function $f_t:\...
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        144 views
        +100

        Structural Stability on Compact $2$-Manifolds with Boundary

        I'm studying the structural stability of vector fields and I'm interested in learning about this phenomenon on compact $2$-manifolds with boundary. Let $M^2$ be a compact connected 2-manifold and $\...
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        votes
        1answer
        97 views

        Symmetries for Julia sets of perturbations of polynomial maps

        This is a naive question. Consider the Julia sets of the map $$ z \mapsto z^n + \lambda / z^k $$ with $z,\lambda \in \mathbb{C}$, and the exponents $n,k \in \mathbb{N}$. For example, for $n=k=3$, ...
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        45 views

        Runge-Kutta 4th order for predator-prey model [on hold]

        I'm trying to compute the numerical solution for a Predator-prey model with 3 equations. This is the model: $$\frac{dx}{dt} =x(1-\frac{x}{k_1})-\frac{pxz}{1+ax+chy}\\ \frac{dy}{dt} =y(1-\frac{y}{k_2})...
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        votes
        0answers
        55 views

        Piecewise linear expanding maps

        Let $(I_{n})$ be a countable infinite disjoint partition of $[0,1)$ into half-open intervals. Let $f:[0,1)\to [0,1)$ be the piecewise linear expanding map with $f(I_{n})=[0,1)$ for all $n$. I suppose ...
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        1answer
        95 views

        Trajectory leaving a set

        Consider the differential equation $\dot{x}=f(x)$, where $f: \mathbb{R}^2 \to \mathbb{R}^2$ is smooth. Given a set $A \subset \mathbb{R}^2$, are there some results saying that whenever $x(0) \in A$, ...
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        54 views

        Nearly eventually almost periodic functions

        Call a function $f: [0, \infty) \to \mathbb R$ nearly eventually almost periodic with period $p > 0$ if for a.e. $x \in [0, p)$, the sequence ${f(x + np)}_{n \in \mathbb N}$ converges. Suppose $f: ...
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        48 views

        Does singularity confinement imply a fixed pattern of irreducible factors?

        Consider a rational map $f \colon (x_1,\ldots,x_n) \mapsto (P_1(x_1,\ldots,x_n),\ldots,P_n(x_1,\ldots,x_n))$, where the $P_i$ are rational functions. Via iteration this map defines a discrete ...
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        votes
        1answer
        78 views

        Discrete dynamical system and bound on norm

        Let $z \in \mathbb R\backslash \left\{2 \right\}$ then I would like to understand the following: Consider the dynamical system with $x_i \in \mathbb C^2:$ $$ x_{i} = \left(\begin{matrix} z &&...
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        votes
        1answer
        333 views

        Formal group law and Koenigs function conjecture?

        Let $f(x,y)$ be a symmetric real function and a formal group law $$G(x + y) = f(G(x),G(y)). \tag{1}$$ Consider the equation $$ h(2x) = f(h(x),h(x)) = A(h(x)). \tag{2}$$ This equation has many ...
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        42 views

        About strange invariant set of the Lozi mappings

        Consider the Tent map: $f_{μ}(x)=μx$, if $x<0.5$ and $f_{μ}(x)=μ(1-x)$ if $x≥0.5$. In this page (https://en.wikipedia.org/wiki/Tent_map) it was stated that: If $μ$ is greater than $2$ the map's ...
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        2answers
        242 views

        Dimension of orbit versus invariant functions

        $\def\CC{\mathbb{C}}$Let $K = \CC(x_1, \ldots, x_n)$ and let $G$ be a countable group of automorphisms of $K$; in the cases I care about, $G \cong \mathbb{Z}$. Then the field of $G$-invariants, $K^G$, ...
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        83 views

        What functional equations are used to extend tetration from the whole numbers to the complex numbers?

        In searching for information regarding extending tetration I have come across half a dozen published papers. But It seems that Abel’s functional equation is at the heart of the extension techniques I’...
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        1answer
        76 views

        A special oscillatory orbit in space

        Edit: According to the comment of Prof. Eremenko I revise the question. 19 years ago, I have heard the following problem from a specialist of dynamical system. During these 19 years, I was in contact ...
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        votes
        1answer
        93 views

        6-periodic billiards trajectory in acute triangle

        We can construct a 3-periodic billiards trajectory in an acute triangle in a classical geometric way, say taking the altitudes. Is there a similar way to construct a 6-periodic billiards?

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