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YCor
YCor
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Continuous-time Extensionextension of a Discrete Dynamical Systemdiscrete dynamical system

It is clear that one can obtain a discrete dynamical system from a continouscontinuous one, but is the converse possible if the system is "nice"?

Define the discrete-time dynamical system on $\mathbb{R}^d$ by $$ x_{n+1} = f(x_n);\, x_0\triangleq x $$ where $f \in C^2(\mathbb{R}^d;\mathbb{R}^d)$ and $x \in \mathbb{R}^d$.

Fix a (large) positive integer $N$, is there a function $F:\mathbb{R}^d\rightarrow \mathbb{R}^d$ such that the solution to the continuous-time dynamical system $$ \partial_t X_t = F(X_t) ; \qquad X_0=x, $$ and $X_n = x_n$ for every $n \in \left\{1,\dots,N\right\}$?

Continuous-time Extension of a Discrete Dynamical System

It is clear that one can obtain a discrete dynamical system from a continous one, but is the converse possible if the system is "nice"?

Define the discrete-time dynamical system on $\mathbb{R}^d$ by $$ x_{n+1} = f(x_n);\, x_0\triangleq x $$ where $f \in C^2(\mathbb{R}^d;\mathbb{R}^d)$ and $x \in \mathbb{R}^d$.

Fix a (large) positive integer $N$, is there a function $F:\mathbb{R}^d\rightarrow \mathbb{R}^d$ such that the solution to the continuous-time dynamical system $$ \partial_t X_t = F(X_t) ; \qquad X_0=x, $$ and $X_n = x_n$ for every $n \in \left\{1,\dots,N\right\}$?

Continuous-time extension of a discrete dynamical system

It is clear that one can obtain a discrete dynamical system from a continuous one, but is the converse possible if the system is "nice"?

Define the discrete-time dynamical system on $\mathbb{R}^d$ by $$ x_{n+1} = f(x_n);\, x_0\triangleq x $$ where $f \in C^2(\mathbb{R}^d;\mathbb{R}^d)$ and $x \in \mathbb{R}^d$.

Fix a (large) positive integer $N$, is there a function $F:\mathbb{R}^d\rightarrow \mathbb{R}^d$ such that the solution to the continuous-time dynamical system $$ \partial_t X_t = F(X_t) ; \qquad X_0=x, $$ and $X_n = x_n$ for every $n \in \left\{1,\dots,N\right\}$?

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ABIM
ABIM
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It is clear that one can obtain a discrete dynamical system from a continous one, but is the converse possible if the system is "nice"?

Define the discrete-time dynamical system on $\mathbb{R}^d$ by $$ x_{n+1} = f(x_n);\, x_0\triangleq x $$ where $f \in C^2(\mathbb{R}^d;\mathbb{R}^d)$ and $x \in \mathbb{R}^d$.

IsFix a (large) positive integer $N$, is there a function $F:\mathbb{R}^d\rightarrow \mathbb{R}^d$ such that the solution to the continuous-time dynamical system $$ \partial_t X_t = F(X_t) ; \qquad X_0=x, $$ and $X_n = x_n$ for every $n \in \mathbb{N}$$n \in \left\{1,\dots,N\right\}$?

It is clear that one can obtain a discrete dynamical system from a continous one, but is the converse possible if the system is "nice"?

Define the discrete-time dynamical system on $\mathbb{R}^d$ by $$ x_{n+1} = f(x_n);\, x_0\triangleq x $$ where $f \in C^2(\mathbb{R}^d;\mathbb{R}^d)$ and $x \in \mathbb{R}^d$.

Is there a function $F:\mathbb{R}^d\rightarrow \mathbb{R}^d$ such that the solution to the continuous-time dynamical system $$ \partial_t X_t = F(X_t) ; \qquad X_0=x, $$ and $X_n = x_n$ for every $n \in \mathbb{N}$?

It is clear that one can obtain a discrete dynamical system from a continous one, but is the converse possible if the system is "nice"?

Define the discrete-time dynamical system on $\mathbb{R}^d$ by $$ x_{n+1} = f(x_n);\, x_0\triangleq x $$ where $f \in C^2(\mathbb{R}^d;\mathbb{R}^d)$ and $x \in \mathbb{R}^d$.

Fix a (large) positive integer $N$, is there a function $F:\mathbb{R}^d\rightarrow \mathbb{R}^d$ such that the solution to the continuous-time dynamical system $$ \partial_t X_t = F(X_t) ; \qquad X_0=x, $$ and $X_n = x_n$ for every $n \in \left\{1,\dots,N\right\}$?

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ABIM
ABIM
  • 5.4k
  • 3
  • 19
  • 41

Continuous-time Extension of a Discrete Dynamical System

It is clear that one can obtain a discrete dynamical system from a continous one, but is the converse possible if the system is "nice"?

Define the discrete-time dynamical system on $\mathbb{R}^d$ by $$ x_{n+1} = f(x_n);\, x_0\triangleq x $$ where $f \in C^2(\mathbb{R}^d;\mathbb{R}^d)$ and $x \in \mathbb{R}^d$.

Is there a function $F:\mathbb{R}^d\rightarrow \mathbb{R}^d$ such that the solution to the continuous-time dynamical system $$ \partial_t X_t = F(X_t) ; \qquad X_0=x, $$ and $X_n = x_n$ for every $n \in \mathbb{N}$?