Determine for the flow of a differential equation

[Askhwhelp]

In each of the following cases, we define a function
:
$\phi$: ${\mathbb R} \times {\mathbb R}^3 \rightarrow {\mathbb R}^3$
. Determine in
each case whether this function could be the flow of a differential equation, and write
down the differential equation.

(a) $\phi_t(\vec{x}) = (8,1,0)$,

(b) $\phi_t(\vec{x}) = \vec{x} \ \text{for all } t,$

(c) $\phi_t(\vec{x}) = \vec{x} + (t,t,t).$

Could anyone helps me to decide how to determine the flow of a differential equation as I have trouble understanding what is the flow of a differential equation, and write down the differential equations?

 

[haruspex]

I'll make a stab at fixing up the Latex, but something still seems to be wrong:

In each of the following cases, we define a function $\phi: {\mathbb R} \times {\mathbb R}^3 \rightarrow {\mathbb R}^3$. Determine in each case whether this function could be the flow of a differential equation, and write down the differential equation.

(a) $\phi_t(\vec{x}) = \vec{x} = (8,1,0)$,

(b) $\phi_t(\vec{x}) = \vec{x} = \vec{x} \ \text{for all } t,$

(c) $\phi_t(\vec{x}) = \vec{x} + (t,t,t)$

Could anyone helps me to decide how to determine the flow of a differential equation as I have trouble understanding what is the flow of a differential equation, and write down the differential equations?

What does $\phi_t(\vec{x}) = \vec{x} = (8,1,0)$ mean? Should it be $\phi_t(\vec{x}) = \vec{x} + (8,1,0)$?
As to what the flow of a differential equation means, try a net search. E.g. http://www.math.sjsu.edu/~simic/Fall05/Math134/flows.pdf.

 

[Askhwhelp]

I'll make a stab at fixing up the Latex, but something still seems to be wrong:

What does $\phi_t(\vec{x}) = \vec{x} = (8,1,0)$ mean? Should it be $\phi_t(\vec{x}) = \vec{x} + (8,1,0)$?
As to what the flow of a differential equation means, try a net search. E.g. http://www.math.sjsu.edu/~simic/Fall05/Math134/flows.pdf.

it should be a) $\phi_t(\vec{x}) = (8,1,0)$ and b) $\phi_t(\vec{x}) = \vec{x}$for all t?

 

[Askhwhelp]

Thank you for the article ... I am still confused by it...Could you show what it means by example?

 

[haruspex]

Thank you for the article ... I am still confused by it...Could you show what it means by example?

I'll try to explain in words.
If you have an ODE dX/dt=F(X), where X is an n-dimensional vector and F is an n-dimensional function of it, then you can think of it as a vector field: at each point X in the space there is a vector F(X) pointing from it. You can imagine starting at some point X₀ in the space and following the chain of vectors for 'time' t. The point we reach is represented as φ_t(X₀). Thus, φ_t is a function which takes the whole space and maps each point to where it would be at time t. Or we can write φ(t,X₀) = φ_t(X₀), making φ a function :$\Re\times\Re^n\rightarrow\Re^n$. This is known as the 'flow'.

 

[Askhwhelp]

I'll try to explain in words.
If you have an ODE dX/dt=F(X), where X is an n-dimensional vector and F is an n-dimensional function of it, then you can think of it as a vector field: at each point X in the space there is a vector F(X) pointing from it. You can imagine starting at some point X₀ in the space and following the chain of vectors for 'time' t. The point we reach is represented as φ_t(X₀). Thus, φ_t is a function which takes the whole space and maps each point to where it would be at time t. Or we can write φ(t,X₀) = φ_t(X₀), making φ a function :$\Re\times\Re^n\rightarrow\Re^n$. This is known as the 'flow'.

i still don't see how dX/dt=F(X) relates to φ(t,X₀) = φ_t? Especially related to setup of my question

 

[pasmith]

In each of the following cases, we define a function
:
$\phi$: ${\mathbb R} \times {\mathbb R}^3 \rightarrow {\mathbb R}^3$
. Determine in
each case whether this function could be the flow of a differential equation, and write
down the differential equation.

(a) $\phi_t(\vec{x}) = (8,1,0)$,

(b) $\phi_t(\vec{x}) = \vec{x} \ \text{for all } t,$

(c) $\phi_t(\vec{x}) = \vec{x} + (t,t,t).$

Could anyone helps me to decide how to determine the flow of a differential equation as I have trouble understanding what is the flow of a differential equation, and write down the differential equations?

A flow on $\mathbb{R}^3$ is a differentiable function $f: \mathbb{R} \times \mathbb{R}^3 \to \mathbb{R}^3$ such that the restrictions $f_t : \mathbb{R}^3 \to \mathbb{R}^3 : x \mapsto f(t,x)$ satisfy:

• $f_0(x) = x$ for all $x \in \mathbb{R}^3$.
• $f_t(f_s(x)) = f_{t+s}(x) = f_s(f_t(x))$ for all $x \in \mathbb{R}^3$ and all $t \in \mathbb{R}$ and all $s \in \mathbb{R}$.

Note that these conditions require $f_0 = f_t \circ f_{-t}$ so that $f_{-t} = f_t^{-1}$ and each $f_t$ is invertible.

To find the differential equation, set $x(t) = \phi_t(x_0)$ for arbitrary constant $x_0$ and differentiate with respect to $t$.