What is Topological Equivalence in Functions and Dynamical Systems?

[marellasunny]

It would be helpful if someone could please explain topological equivalence of functions in simple words?
I am working on dynamical systems and chaos theory.In the underlying material,topological equivalence has taken a more complex definition involving orbits.Please be kind enough to explain this part also.
'What does it mean for 2 maps to be topologically equivalent'?
Thankyou.

 

[K41]

The dynamics of one system A are topologically equivalent to the dynamics of another system B, if a homeomorphic function exists such that it can map from one vector A to the other vector B and preserve the direction of time.

 

[marellasunny]

Is this the same definition of 'holomorphic function' we use in complex analysis?i.e A function f is holomorphic at a point z if it is C differentiable in a neighborhood of z/preserves angles and orientation in space.??Does preserving the direction of time also come into special consideration only for dynamical systems?

Are the topological properties of a dynamical system the same as the properties of a topological space?

What does it mean when one says the perturbed system is topologically isomorphic to the unpurturbed system?

 

[HallsofIvy]

No one has mentioned "holomorphic". djpailo used the word "homeomorphic".

A "homeomorphism" is an invertible, continuous, function from one topological space to another and two topological spaces are "equivalent" if and only if there exist a homeomophism from one to the other.

 

[CellCoree]

Topological equivalence refers to the idea that two functions or maps have the same overall shape or structure, even if their specific values or coordinates may differ. This means that the general behavior or patterns of the functions are the same, but the details may vary. In other words, the functions have the same topological properties, such as continuity and connectedness.

In the context of dynamical systems and chaos theory, topological equivalence means that two systems have the same overall dynamics, even if their specific trajectories or orbits may differ. This can be seen by looking at the overall behavior of the systems, such as the presence of attractors or the stability of the system, rather than focusing on individual trajectories.

To determine whether two functions or systems are topologically equivalent, we look at their overall structure and behavior rather than their specific values or coordinates. This allows us to understand the general behavior and patterns of the functions or systems, which can be useful in studying complex systems and phenomena.