the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 30 Oct 2025
On transversality and the characterization of finite time hyperbolic subspaces in chaotic attractors
Abstract. We examine the local stable and unstable manifolds of chaotic attractors and their associated growth rates for the quantification of (non-)hyperbolicity in low dimensional nonlinear autonomous dissipative models. This is motivated by a desire for a deeper understanding of transversality and hyperbolicity to inform key challenges to prediction in spatially extended chaotic systems in geophysical flows. In particular, we apply local measures of chaos to quantify the relationship between transversality, dimension, and hyperbolicity on the subspaces of the attractors' invariant manifolds. We consider unstable directions and growth rates determined over finite time intervals, specifically those predicated on information over the past evolution i.e., finite time backwards Lyapunov vectors, and those that include information from both the past and future i.e., finite time covariant Lyapunov vectors. Our study reveals general properties across a diverse set of chaotic attractors at short, intermediate and extended forecast horizons associated with the emergence of distinct locally evolving regions of instability.
- Preprint
(42957 KB) - Metadata XML
- BibTeX
- EndNote
Status: closed
-
RC1: 'Comment on egusphere-2025-5208', Anonymous Referee #1, 27 Nov 2025
-
AC1: 'Reply on RC1', Terence O'Kane, 14 Jan 2026
We thank the the reviewer for their efforts and appreciate the favourable comments.
With regard to the two points raised: the suggestion to discuss application prospects of the approach in the Earth system is a very good one which we will certainly address in the revised manuscript. Briefly, as most approaches in operational ensemble prediction systems are posed in terms of projection of the dominant growing error mode onto the leading Lyapunov vector, our work shows that over finite windows - as is the case in weather prediction - error growth can be more complex and may be more accurately described by the local Kaplan-Yorke dimension. This observation goes directly to the reviewers second comment that generalization to the Earth system i.e., a multi-scale system, does indeed reflect the mutual influence between different scales as we see in the chaotic attractors with dimension greater than 3. However, our recent work has shown that within high dimensional multi-scale systems, the emergence of persistent coherent features such as atmospheric blocking, is in fact often associated with the appearance of transient local low dimensional attractors.
We will include additional discussion of these points in the revision.
Citation: https://doi.org/10.5194/egusphere-2025-5208-AC1
-
AC1: 'Reply on RC1', Terence O'Kane, 14 Jan 2026
-
RC2: 'Comment on egusphere-2025-5208', Anonymous Referee #2, 18 Dec 2025
-
AC2: 'Reply on RC2', Terence O'Kane, 14 Jan 2026
We thank the reviewer for their careful consideration of our submission.
We address each of their comments point by point below:
1. The analysis is primarily conducted on low-dimensional chaotic models (Eg. Lorenz 63 or a 9-variable model). It is well known that low-dimensional models lack the multi-scale interactions and spatial extensiveness characteristic of high-dimensional turbulent systems. The authors should provide a more in-depth discussion on the validity of extrapolating local hyperbolicity and transversality features observed in these simple attractors to high-dimensional systems.
Authors: Clearly as the dimensionality increases so does the importance of multi-scale interactions. However, where we are concerned with emergent properties such as coherent features, often it is the appearance of a local low dimensional slow manifold that determines the lifecycle of the structure. This is where the analogy to local hyperbolicity and transversality features as described in the current work becomes relevant. We completely agree this is a key point for further more in-depth discussion.
2. The introduction reviews previous work, such as Quinn et al. (2020), regarding the finite-time Kaplan-Yorke dimension (dimKY) and data assimilation. Given the overlap in the author list and the thematic similarity, it is essential to clearly distinguish the contributions of the current manuscript from these prior studies. Does the novelty lie in the introduction of a new analytical perspective or in understanding the underlying dynamical mechanisms? I recommend explicitly stating the innovations of this work in the Introduction or Discussion.
Authors: Agreed. We will clearly identify the novel aspects of the current work and differentiate these from our previous work on data assimilation in the revised manuscript.
3. The paper would benefit significantly from a stronger connection to physical mechanisms. Specifically, when the system exhibits some properties, such as “non-hyperbolicity”, what does these properties correspond to in terms of the physical state or dynamical events within the model?.
Authors: Thanks - this point was also raised by the other reviewer. There is much to discuss in terms of how ideas of tranversality and hyperbolicity of the described chaotic attractors inform on the characteristics and diversity of specific manifestations of persistent local coherent structures within the context of higher dimensional multi-scale systems.
4. Should the second column of Figure 1 be labeled as “FTBLE2” ?
Authors: yes - thanks!
Citation: https://doi.org/10.5194/egusphere-2025-5208-AC2
-
AC2: 'Reply on RC2', Terence O'Kane, 14 Jan 2026
Status: closed
-
RC1: 'Comment on egusphere-2025-5208', Anonymous Referee #1, 27 Nov 2025
This study examine the local stable and unstable manifolds of chaotic attractors and their associated growth rates for the quantification of (non-)hyperbolicity in low dimensional nonlinear autonomous dissipative models. I believe the manuscript fully meets the journal's requirements. There are some Minor comments:
- Suggest discussing the application prospects in the Earth system.
- The Earth system is multiple scalessystem, does the mutual influence between different scales affect the results?
Citation: https://doi.org/10.5194/egusphere-2025-5208-RC1 -
AC1: 'Reply on RC1', Terence O'Kane, 14 Jan 2026
We thank the the reviewer for their efforts and appreciate the favourable comments.
With regard to the two points raised: the suggestion to discuss application prospects of the approach in the Earth system is a very good one which we will certainly address in the revised manuscript. Briefly, as most approaches in operational ensemble prediction systems are posed in terms of projection of the dominant growing error mode onto the leading Lyapunov vector, our work shows that over finite windows - as is the case in weather prediction - error growth can be more complex and may be more accurately described by the local Kaplan-Yorke dimension. This observation goes directly to the reviewers second comment that generalization to the Earth system i.e., a multi-scale system, does indeed reflect the mutual influence between different scales as we see in the chaotic attractors with dimension greater than 3. However, our recent work has shown that within high dimensional multi-scale systems, the emergence of persistent coherent features such as atmospheric blocking, is in fact often associated with the appearance of transient local low dimensional attractors.
We will include additional discussion of these points in the revision.
Citation: https://doi.org/10.5194/egusphere-2025-5208-AC1
-
RC2: 'Comment on egusphere-2025-5208', Anonymous Referee #2, 18 Dec 2025
-
AC2: 'Reply on RC2', Terence O'Kane, 14 Jan 2026
We thank the reviewer for their careful consideration of our submission.
We address each of their comments point by point below:
1. The analysis is primarily conducted on low-dimensional chaotic models (Eg. Lorenz 63 or a 9-variable model). It is well known that low-dimensional models lack the multi-scale interactions and spatial extensiveness characteristic of high-dimensional turbulent systems. The authors should provide a more in-depth discussion on the validity of extrapolating local hyperbolicity and transversality features observed in these simple attractors to high-dimensional systems.
Authors: Clearly as the dimensionality increases so does the importance of multi-scale interactions. However, where we are concerned with emergent properties such as coherent features, often it is the appearance of a local low dimensional slow manifold that determines the lifecycle of the structure. This is where the analogy to local hyperbolicity and transversality features as described in the current work becomes relevant. We completely agree this is a key point for further more in-depth discussion.
2. The introduction reviews previous work, such as Quinn et al. (2020), regarding the finite-time Kaplan-Yorke dimension (dimKY) and data assimilation. Given the overlap in the author list and the thematic similarity, it is essential to clearly distinguish the contributions of the current manuscript from these prior studies. Does the novelty lie in the introduction of a new analytical perspective or in understanding the underlying dynamical mechanisms? I recommend explicitly stating the innovations of this work in the Introduction or Discussion.
Authors: Agreed. We will clearly identify the novel aspects of the current work and differentiate these from our previous work on data assimilation in the revised manuscript.
3. The paper would benefit significantly from a stronger connection to physical mechanisms. Specifically, when the system exhibits some properties, such as “non-hyperbolicity”, what does these properties correspond to in terms of the physical state or dynamical events within the model?.
Authors: Thanks - this point was also raised by the other reviewer. There is much to discuss in terms of how ideas of tranversality and hyperbolicity of the described chaotic attractors inform on the characteristics and diversity of specific manifestations of persistent local coherent structures within the context of higher dimensional multi-scale systems.
4. Should the second column of Figure 1 be labeled as “FTBLE2” ?
Authors: yes - thanks!
Citation: https://doi.org/10.5194/egusphere-2025-5208-AC2
-
AC2: 'Reply on RC2', Terence O'Kane, 14 Jan 2026
Viewed
| HTML | XML | Total | BibTeX | EndNote | |
|---|---|---|---|---|---|
| 211 | 152 | 35 | 398 | 74 | 66 |
- HTML: 211
- PDF: 152
- XML: 35
- Total: 398
- BibTeX: 74
- EndNote: 66
Viewed (geographical distribution)
| Country | # | Views | % |
|---|
| Total: | 0 |
| HTML: | 0 |
| PDF: | 0 |
| XML: | 0 |
- 1
This study examine the local stable and unstable manifolds of chaotic attractors and their associated growth rates for the quantification of (non-)hyperbolicity in low dimensional nonlinear autonomous dissipative models. I believe the manuscript fully meets the journal's requirements. There are some Minor comments: