Dear colleagues
To find coexisting attractors in a chaotic system, I use the continuation diagram. Here in each iteration, the initial conditions x(0) for the chaotic system are set as the final conditions x(t_final) from the previous simulation.
We do so as we increase the parameter under study (forward continuation diagram), and as we decrease the parameter (backward continuation diagram).
In a system I am studying though, I still know that coexisting attractors exist, and using both continuation diagrams, I still cannot depict all of them. The diagram cannot 'catch' them.
Is there an alternative, or a solution to this?
Continuation diagrams are used to visualize the behavior of dynamical systems as parameters change. Coexisting attractors in such diagrams indicate the presence of multiple stable states or patterns in the system. These could be different attractors such as fixed points, periodic orbits, or chaotic behavior, existing simultaneously under varying conditions. Identifying and analyzing these coexisting attractors can provide insights into the complex behavior of the system.
Dear Lazaros Moysis
Given the wording of the question, it seems that you are already aware of the existence of coexisting states and are looking for a way to visualize them. Calculating a continuation diagram restricts the state space to be studied. I would suggest a bifurcation diagram with random changes in initial conditions, i.e. use at least 20 different and random initial conditions for each bifurcation parameter (you would need to define the possible range of these conditions). Once you have done this, the dynamical richness of the system should become apparent, allowing you to create new diagrams with more initial conditions (or in larger ranges) for specific values of the bifurcation parameter, leading to the identification of a region of interest for which you can calculate the associated basin of attraction.
If there is something you think I can support, feel free to write me
Best regards
Coexisting attractors can be identified by tracking parameter changes across the continuation diagram, revealing where these attractors coexist and potentially interact. This analysis helps understand the complexity of a system's dynamics. If you need assistance with interpreting specific continuation diagrams or attractor behavior, please give more details.
I have prepared a video on finding coexisting attractors in the Lorenz chaotic system https://www.youtube.com/watch?v=vXOK45HbvBU
Code for the bifurcation diagram is available here https://uk.mathworks.com/matlabcentral/fileexchange/156752-bifurcation-diagram-for-the-lorenz-chaotic-system?s_tid=srchtitle
- Badges
- Science topic