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Rigorously proven chaos in chemical kinetics

M. Susits, J. Tóth

Abstract

This study addresses a longstanding question regarding the mathematical proof of chaotic behavior in kinetic differential equations. Following the numerous numerical and experimental results in the past 50 years, we introduce two formal chemical reactions that rigorously demonstrate this behavior. Our approach involves transforming chaotic equations into kinetic differential equations and subsequently realizing these equations through formal chemical reactions. The findings present a novel perspective on chaotic dynamics within chemical kinetics, thereby resolving a longstanding open problem.

Rigorously proven chaos in chemical kinetics

Abstract

This study addresses a longstanding question regarding the mathematical proof of chaotic behavior in kinetic differential equations. Following the numerous numerical and experimental results in the past 50 years, we introduce two formal chemical reactions that rigorously demonstrate this behavior. Our approach involves transforming chaotic equations into kinetic differential equations and subsequently realizing these equations through formal chemical reactions. The findings present a novel perspective on chaotic dynamics within chemical kinetics, thereby resolving a longstanding open problem.
Paper Structure (14 sections, 2 theorems, 25 equations, 9 figures, 2 tables)

This paper contains 14 sections, 2 theorems, 25 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Chaos in chemical kinetic experiments and models
  3. Rigorously proven chaos in formal kinetic models
  4. The Lorenz reaction
  5. Trapping region of the Lorenz system
  6. Transforming the Lorenz system
  7. A realization of the transformed Lorenz system
  8. The Chen reaction
  9. Trapping region of the Chen system
  10. The transformed Chen system and its realization
  11. Discussion and outlook
  12. Appendix
  13. Induced kinetic differential equations
  14. Characteristics of chaotic behavior

Key Result

Proposition 1

For any $Z_0 \leq 0$ and $U_0 \geq U_{0,\min}$ consider the ellipsoids $i=0,1,2,\dots$ where for $i \geq 1$ with $B>A/(A-Z_0)$. In the upper half-space $H_i:Z>Z_i$ let $\Omega_i=E^V_i\cap H_i$ where $E^V_i$ is the region embraced by $E_i$. Moreover for $N\geq 0$ define Then, there exists a non-negative integer $N$ such that $\Omega_{0,1,\dots,N}$ is a trapping region. The value of $N$ is determi

Figures (9)

  • Figure 1: $D: 28x^2 + y^2 + \frac{8}{3}(z-28)^2 = \frac{6272}{3}$ and $E : 28x^2 + 10 y^2 + 10(z - 56)^2 = 200^2$
  • Figure 2: Time evolution of the first component in the Lorenz equation with the parameters as given in the text (left) and the corresponding trajectory (right) started from $(1,2,3)$ on the time interval 0-50.
  • Figure 3: Time evolution of the first component in the transformed Lorenz reaction (left) and the corresponding trajectory (right) started from $(101,102,13)$ on the time interval 0-0.00012.
  • Figure 4: The Feinberg--Horn--Jackson graph of the reaction \ref{['eq:lorenz01']}--\ref{['eq:lorenz06']}.
  • Figure 5: Time evolution of the first component in the Chen equation with the parameters as given in the text (left) and the corresponding trajectory (right) started from $(3,1,4)$ on the time interval 0-13.
  • ...and 4 more figures

Theorems & Definitions (10)

  • Proposition 1
  • Lemma 1
  • Definition 1: Canonic realization
  • Definition 2: Devaney
  • Definition 3
  • Definition 4: Smale, horseshoe
  • Definition 5: Smith, topological entropy
  • Definition 6
  • Definition 7: Physics, Lyapunov
  • Definition 8: Period doubling