# Is each path in a phase portrait unique?

• I
• crastinus
In summary, the conversation discusses the concept of phase space and phase portrait, with the former being the space of all possible states and the latter being the phase space with connected paths following the system's rule of transformation. For a classical system, there is only one path in the phase portrait between two states over a fixed time interval. To understand this concept, the conversation suggests using theorems and provides one for initial value problems with a smooth vector field. The theorem states that for any bounded open domain, there exists a constant time interval where the solution is defined and the mapping between the domain and its transformed state is a diffeomorphism.

#### crastinus

As I understand it, the phase space is the space of all possible states, without reference to the rule of transformation from one state to another; whereas the phase portrait is the phase space with the paths from one state to another connected following the system's rule of transformation.

If that is so, then, for a classical system, is there only one path in the phase portrait between two states A and B over a fixed time interval from ti to tj?

I believe that the best way to grasp such things is to use theorems.
Consider an initial value problem $$\dot x=v(x),\quad x(0)=\hat x,\quad x=(x^1,\ldots,x^m)\in\mathbb{R}^m.\qquad (*)$$ We also assume for simplicity that the vector field ##v## is smooth in ##\mathbb{R}^m##. This assumption is sufficient to guarantee existence and uniqueness of the solution by the Cauchy theorem. The phase space is ##\mathbb{R}^m##.
Let ##g^t(\hat x)## stand for solution to problem (*)

Theorem. Let ##K\subset \mathbb{R}^m## be any bounded open domain. Then there exists a positive constant ##T## such that the solution ##g^t(\hat x)## is defined for all ##t\in[-T,T]## and for all ##\hat x\in K##. Moreover (this is the answer to your question) for any ##t\in[-T,T]## the mapping ##\hat x\mapsto g^t(\hat x)## is a diffeomorphism between ##K## and ##g^t(K)##.
If ##t,s,t+s\in[-T,T]##then ##g^{t+s}(\hat x)=g^t(g^s(\hat x))##

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• vanhees71
This is great. Just what I need! I confess that I'm not sure I understand it yet, though. Let me work on it a bit before replying.

Thanks again!

## 1. What is a phase portrait?

A phase portrait is a graphical representation of the qualitative behavior of a dynamical system. It shows the trajectories, or paths, that the system can take as it evolves over time.

## 2. How does a phase portrait help in understanding a dynamical system?

A phase portrait provides a visual representation of the behavior of a dynamical system, making it easier to understand and analyze. It can reveal important features such as stability, periodicity, and attractors.

## 3. Are all paths in a phase portrait unique?

No, not all paths in a phase portrait are unique. Some paths may overlap or intersect with each other, while others may follow the same trajectory but in reverse direction.

## 4. How can we determine if a path in a phase portrait is unique?

The uniqueness of a path in a phase portrait depends on the specific conditions and parameters of the dynamical system. To determine if a path is unique, we can analyze the equations that govern the system and look for solutions that yield the same trajectory.

## 5. What factors can affect the uniqueness of paths in a phase portrait?

The uniqueness of paths in a phase portrait can be influenced by the initial conditions, parameters of the system, and the behavior of the system itself. Nonlinear systems, in particular, can exhibit complex and non-unique paths in their phase portraits.