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Is the initial value problem of an ODE considered as a dynamic system?

A dynamic system is defined as

In the most general sense, a dynamical system is a tuple (T, M, Φ) where T is a monoid, written additively, M is a set and Φ is a function $$ \Phi: U \subset T \times M \to M $$ with $$ I(x) = \{ t \in T : (t,x) \in U \}\,$$ $$ \Phi(0,x) = x\,$$ $$ \Phi(t_2,\Phi(t_1,x)) = \Phi(t_1 + t_2, x),\, for \, t_1, t_2, t_1 + t_2 \in I(x)\, $$

For an initial value problem $$ \dot{\boldsymbol{x}}=\boldsymbol{v}(t,\boldsymbol{x}) $$ $$ \boldsymbol{x}|_{{t=0}}=\boldsymbol{x}_0 $$

the solution is an evolution function $$ \boldsymbol{{x}}(t)=\Phi(t,\boldsymbol{{x}}_0) $$

I don't think $\Phi$ satisfy $\Phi(t_2,\Phi(t_1,x)) = \Phi(t_1 + t_2, x),\, for \, t_1, t_2, t_1 + t_2 \in I(x)\, $. For example, when $v(t,x) = f(t)$, $\Phi(t,\boldsymbol{{x}}_0) = \int_0^t f(s) ds + x_0$.

But I often heard of the initial value problem of ODE and dynamic system together. If the former is not an example of the latter, I wonder why they are mentioned together?


There is also a related question posted before here Construction of dynamical systems from ODEs with initial values.

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up vote 3 down vote accepted

Your guess that the flow map $\Phi(t,x_0) =x(t)$ in the ODE with initial point $x_0$ satisfies the dynamical system property is actually correct, if the ODE is autonomous (time-independent), and if solutions exist locally in time. In your example, $\dot{x} = f(t)$ depends on time, so at face value the flow map does not satisfy the semigroup property. However, if $\dot{x} = v(x,t)$ is time-dependent, then we can relabel $t$ to a new variable $y$ and form the system $$ \dot{x} = v(x,y), \, \dot{y} = 1$$ and this can be interpreted as a time-independent system.

Edit: To answer the questions posed in the comments:

  1. An ODE of the form $\dot{x} = v(x)$ does not necessarily even have a solution; the existence and uniqueness of such solutions, which is one of the main items proven in a course on the subject, is required for the idea of an ODE being a dynamical system to even make sense. A sufficient condition for existence/uniqueness is that the vector field $v(x)$ is a locally Lipschitz continuous map.

  2. From this, the semigroup property for the flow map $\Phi$ can be demonstrated. The flow $\Phi(t,x_0)$ gives the value of $x(t)$ as the solution to the initial value problem $x(0) = x_0$. Now, $\Phi(t_2, \Phi(t_1, x_0))$ gives the unique value after $t_2$ time after the initial condition given by $x(0) = \Phi(t_1, x_0)$. In turn, $\Phi(t_1, x_0)$ is the unique value $x(t_1)$ to the initial condition given by $x(0) = x_0$. Then $\Phi(t_1 + t_2, x_0)$ gives the solution to up to time $t_1 + t_2$; by uniqueness the corresponding solution curve $x(t)$ passes through the point $x(t_1)$ and $x(t_1 + t_2)$. Then, by uniqueness, $x(t_1 + t_2)$ must also be the unique value which is $t_2$-time after $x(t_1)$; therefore we have $$ \Phi(t_1 + t_2, x_0) = \Phi(t_2, \Phi(t_1, x_0))$$

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Thanks! But isn't it $Φ(t_2,Φ(t_1,x_0))=∫^{t_2}_0 f(s)ds+Φ(t_1,x_0) = ∫^{t_2}_0 f(s)ds+ ∫^{t_1}_0 f(s)ds + x_0$? – Tim Nov 18 '12 at 21:49
That's a good catch, my answer will be edited shortly. – Christopher A. Wong Nov 18 '12 at 21:54
Thanks for the edit! (1) What does "solutions exist locally in time" mean? (2) Can you show a little bit that the initial problem of ODE is a dynamic system when the ODE is autonomous (time-independent), and if solutions exist locally in time? – Tim Nov 19 '12 at 0:00
Some references for my questions are also appreciated. Thanks! – Tim Nov 19 '12 at 1:58

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