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While modern physics and biology satisfactorily explain the passage from the Big Bang to the formation of Earth and the first cells to present-day life, respectively, the origins of biochemical life still remain an open question. Since life, as we know it, requires extremely long genetic polymers, any answer to the question must explain how an evolving system of polymers of ever-increasing length could come about on a planet that otherwise consisted only of small molecular building blocks. In this work we show that, under realistic constraints, an abstract polymer model can exhibit dynamics such that attractors in the polymer population space with a higher average polymer length are also more probable. We generalize from the model and formalize the notions of and for chemical reaction networks with multiple attractors. The complexity of a species is defined as the minimum number of reactions needed to produce it from a set of building blocks, which in turn is used to define a measure of complexity for an attractor. A transition between attractors is considered to be a if the attractor with the higher probability also has a higher complexity. In an environment where only monomers are readily available, the attractor with a higher average polymer length is more complex. Thus, by this criterion, our abstract polymer model can exhibit progressive evolution for a range of thermodynamically plausible rate constants. We also formalize criteria for and evolution and explain the role of autocatalysis in obtaining them. Our work provides a basis for searching for prebiotically plausible scenarios in which long polymers can emerge and yield populations with even longer polymers. Additionally, the existence of features like history dependence and open endedness support the view that the path from chemistry to biology was one of gradual complexification rather than an instantaneous origin of life. Published by the American Physical Society2025
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The hidden complexity of a double-scroll attractor: Analytic proofs from a piecewise-smooth system
Double-scroll attractors are one of the pillars of modern chaos theory. However, rigorous computer-free analysis of their existence and global structure is often elusive. Here, we address this fundamental problem by constructing an analytically tractable piecewise-smooth system with a double-scroll attractor. We derive a Poincaré return map to prove the existence of the double-scroll attractor and explicitly characterize its global dynamical properties. In particular, we reveal a hidden set of countably many saddle orbits associated with infinite-period Smale horseshoes. These complex hyperbolic sets emerge from an ordered iterative process that yields sequential intersections between different horseshoes and their preimages. This novel distinctive feature differs from the classical Smale horseshoes, directly intersecting with their own preimages. Our global analysis suggests that the structure of the classical Chua attractor and other figure-eight attractors might be more complex than previously thought.
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- PAR ID:
- 10406179
- Publisher / Repository:
- American Institute of Physics
- Date Published:
- Journal Name:
- Chaos: An Interdisciplinary Journal of Nonlinear Science
- Volume:
- 33
- Issue:
- 4
- ISSN:
- 1054-1500
- Page Range / eLocation ID:
- Article No. 043119
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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