Hi uttam
For the attractors to be exist in phase space ,it has to be a contraction & that is formalized in "Liovelle's theorem".Attractors exist only in dissipative systems!
and often ,not always Attractors are fractals and in order to maintain the fractal structure (i.e structure at all scales ) , it has to disspitate energy in self similar way.
Dear Uttam,
we all have always to be a little bit careful with the definitions.
1) If you choose to define "chaotic behaviour" as the presence of a "strange attractor", you should define the term "strange attractor" first. Else, confusions are likely to arise..
2) If you define the term "strange attractor" and choose to identify them with "chaotic behaviour", that's ok, but you miss a big and very intetersting category of systems with "complex behaviour" (geodesic flows on manifolds with negative curvature, billiards, hamiltonian chaos, etc), having no "strange attractors".
3) Precise definitions of "chaos", depending on the viewpoint of the researcher, refer to well defined concepts, such as topological entropy, or denseness of the set of periodic orbits in a compact subset of the phase space+a dense non periodic orbit. If you wish to further study these concepts, may I propose the book "Introduction to the modern theory of dynamical systems", of Katok and Hasselblatt? You will find there all the details necessary to distinguish between different types of behaviour.
4) Concerning your other question, I am having a problem in understanding it. What is the difference between "phase space trajectory" and "real trajectory" ?
Hope I helped,
Stavros.
Hi Uttam... Chaotic orbits (largest lyapunov exponent is positive) are called as strange attarctor. Even a strange non chaotic attractors (SNA) also exists where the lyapunov exponent is non-positive. For complete definition and characterization of SNA , please check the paper of Ulrike Feudel and Arkady Pikovsky . There is another paper in arxiv version written by Awadesh Prasad, Surendra Singh Negi, and Ramakrishna Ramaswamy.
Here is the links:
1.http://scitation.aip.org/content/aip/journal/chaos/5/1/10.1063/1.166074
2.http://arxiv.org/pdf/nlin/0105022.pdf
Thank you, Anastassiou and Hens. Now, I realize that strangeness and chaoticity are two separate things. When there is chaoticity, there are strange attractors. But opposite is not true. So one has to introduce strange nonchaotic attractor. Concerning my second questions, by phase space I meant position-momentum (or position-velocity) space and by real trajectory I meant only positions.
Hello Uttam,
Concerning your second question: I have read that strange attractors in phase space are fractal. Although the phase space trajectories are fractal the real trajectories (I mean, positions with time) should not necessarily be fractal. Am I right?
A strange attractor (also called a chaotic attractor) has a fractal dimension in phase space. The second part of the question has no sense at all.
Best Regards,
Miguel AF Sanjuan
Thanks Sanjuan. I got your point that a strange attractor has a fractal dimension in phase space. Particles under random walk motion can give rise to a fractal pattern, e.g., diffusion-limited aggregation. My point was whether it is possible that a system of particles whose motion is constrained by some strange attractor will also give rise to a fractal structure.
Hi Uttam,
First of all, a strange attractor is an attractor. This means that a dynamical system evolves asymptotically to this attractor, which is not a fixed point, a limit cycle or a quasiperiodic orbit. I guess your question is not well posed. What do you mean by a motion constrained by an strange attractor? The strange attractor possesses a fractal structure in phase space by definition.
Miguel
Hi uttam
For the attractors to be exist in phase space ,it has to be a contraction & that is formalized in "Liovelle's theorem".Attractors exist only in dissipative systems!
and often ,not always Attractors are fractals and in order to maintain the fractal structure (i.e structure at all scales ) , it has to disspitate energy in self similar way.
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