In the ideal noiseless case (with infinite observation time) the attractors reconstructed with different delay values should still be diffeomorphic to the original attractor, as long as the embedding dimension satisfies the conditions of the Takens’ theorem. AFAIK, the choice of delay affects the geometry of the embedding but does not break the diffeomorphic relationship.
However, in noisy cases (i.e., the real world) the diffeomorphism guarantee no longer holds, and inappropriate delay values can significantly distort the attractor’s structure. Even in the noiseless case, very small or very large delays can lead to reconstructions that are mathematically valid but less useful for practical analysis. I have shown the example for increasing delay where the geometry of the attractor is too unfolded: using that geometry for predicting the next state will surely lead to a poor estimate.
Regarding the 2D projections, you are right: they do not preserve the diffeomorphism to the original attractor and are just a way to grasp the higher-dimensional structure. I do not think that someone uses the 2D projection beyond visual inspection, in fact.
In a nutshell, I could say that even if the math allows you, theoretically, to reconstruct the attractor (under the relatively strict requirements of the Takens' theorem), it hardly guarantees that it could be useful for your applications.
Thank you very much! Really nice post! I have only one small question. Can it be that, only in the noiseless case, attractors reconstructed for different delay values (1, 5, 20, 50, 100 200) are still diffeomorphic to the original one? You use m=6, so an embedding space is 6-dimensional, however, you show only 2d projections of it.
Of course, the fact of being merely diffeomorphic does not make such projections useful. However, as I understand the formulation of Takens theorem, such embeddings should be still diffeomorphic regardless of the choice of the delay in an ideal noiseless case and infinite observation time.
I apologise in advance if it is a stupid question.
Thanks for the kind feedback.
In the ideal noiseless case (with infinite observation time) the attractors reconstructed with different delay values should still be diffeomorphic to the original attractor, as long as the embedding dimension satisfies the conditions of the Takens’ theorem. AFAIK, the choice of delay affects the geometry of the embedding but does not break the diffeomorphic relationship.
However, in noisy cases (i.e., the real world) the diffeomorphism guarantee no longer holds, and inappropriate delay values can significantly distort the attractor’s structure. Even in the noiseless case, very small or very large delays can lead to reconstructions that are mathematically valid but less useful for practical analysis. I have shown the example for increasing delay where the geometry of the attractor is too unfolded: using that geometry for predicting the next state will surely lead to a poor estimate.
Regarding the 2D projections, you are right: they do not preserve the diffeomorphism to the original attractor and are just a way to grasp the higher-dimensional structure. I do not think that someone uses the 2D projection beyond visual inspection, in fact.
In a nutshell, I could say that even if the math allows you, theoretically, to reconstruct the attractor (under the relatively strict requirements of the Takens' theorem), it hardly guarantees that it could be useful for your applications.
Thank you very much! Really nice post! I have only one small question. Can it be that, only in the noiseless case, attractors reconstructed for different delay values (1, 5, 20, 50, 100 200) are still diffeomorphic to the original one? You use m=6, so an embedding space is 6-dimensional, however, you show only 2d projections of it.
Of course, the fact of being merely diffeomorphic does not make such projections useful. However, as I understand the formulation of Takens theorem, such embeddings should be still diffeomorphic regardless of the choice of the delay in an ideal noiseless case and infinite observation time.
I apologise in advance if it is a stupid question.