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Differences and similarities in the analysis of Lorenz, Chen, and Lu systems
G. A. Leonov, N. V. Kuznetsov
TL;DR
The paper addresses whether transformations make Lorenz-like systems equivalent and permit universal treatment of their behavior. It compares Lorenz, Chen, and Lu analyses, emphasizing time reversal and negative parameters. It reports obstacles to standard analysis and identifies an unresolved conjecture about attractor dimension for classical Chen and Lu parameters.
Problem
The paper examines the actively discussed question of whether transformations make Lorenz-like systems equivalent and allow their behavior to be considered universally.
Source: §1 · p. 1Method
The paper compares the Lorenz, Chen, and Lu systems, focusing on analytical and numerical difficulties associated with time inversion and negative parameters.
Source: §6 · p. 16Results
Time-inverting reductions obstruct effective attractor visualization, dimension analysis, and homoclinic proofs; the paper also states that a set cannot be positively invariant or dissipative in Levinson's sense.
Source: §6 · pp. 16–17Takeaways & Limitations
The analysis identifies a need for effective analytical methods to study chaotic behavior and attractor characteristics under time inversion or negative parameters.
Source: §6 · p. 16For classical Chen and Lu parameters with d=−1, coincidence of attractor Lyapunov dimension with its value at the zero stationary point remains a conjecture requiring new Lyapunov functions and careful analysis.
Source: §4.2 · p. 14
Abstract
from arXiv · showhide
Currently it is being actively discussed the question of the equivalence of various Lorenz-like systems and the possibility of universal consideration of their behavior, in view of the possibility of reduction of such systems to the same form with the help of various transformations. In the present paper the differences and similarities in the analysis of the Lorenz, the Chen and the Lu systems are discussed and it is shown that the Chen and the Lu systems are valuable for the development of new methods for the analysis of chaotic systems.
1. Introduction
The paper addresses debate over whether transformations make Lorenz-like systems equivalent and their behavior universally treatable. It compares the Lorenz, Chen, and Lu systems and argues that Chen and Lu are valuable for developing methods to analyze chaotic systems.
1. Introduction
- The equivalence debate asks whether transformations can reduce Lorenz-like systems to a common form for universal consideration of their behavior.The introduction notes that this question is actively discussed.Source: §1 · p. 1
- A cited claim holds that Chen and Lu are generically only particular cases of the Lorenz system.The paper quotes this claim while setting out the debate.Source: §1 · pp. 1–2
- The paper discusses differences and similarities among the systems and presents Chen and Lu as valuable for developing new methods to analyze chaotic systems.Source: §1 · p. 2
2. Lorenz-like systems: Lorenz, Chen, Lu, and Tigan systems
The section introduces the Lorenz, Chen, and Lu equations, transformations relating them, and parameter cases relevant to their dynamics. It also motivates their study by noting that Chen and Lu have transition scenarios similar to Lorenz but use fewer parameters.
2. Lorenz-like systems: Lorenz, Chen, Lu, and Tigan systems
- The Lorenz, Chen, and Lu systems are quadratic three-variable Lorenz-like systems, with distinct linear terms in their y equations.The Chen equation includes (c−a)x+cy, whereas the Lu equation includes cy; the Lorenz form uses ρx−y.Source: §2 · p. 2
- The h=a transformation was proposed for the Chen and Lu systems, while later work used h=−c to reduce them to Lorenz form.For positive chaotic parameters, h=−c reverses time; when c=0, the systems do not become linear and their dynamics may remain of interest.Source: §2 · pp. 2–3
- For c=0, the Chen and Lu systems are described as particular cases of the T-system, with an additional transformation used to reach that form.Source: §2 · pp. 3–4
- For Lorenz parameters {σ, β, ρ}={10, 8/3, 28}, a chaotic strange attractor was found numerically after the fixed points lose stability as ρ increases.For 1<ρ<24.74, two additional stable fixed points appear; for ρ>24.74, all three fixed points are unstable.Source: §2 · p. 4
- Chen and Lu are chaotic at the cited parameter sets {a, b, c}={35, 3, 28} and {36, 3, 20}, respectively, but may not be chaotic for other parameters.Source: §2 · p. 4
- A generalized system contains the Lorenz, Chen, and Lu systems as special cases, with their parameter assignments distinguished by d and r.The passage gives d=1 for Lorenz, d=−c and r=c−a for Chen, and d=−c and r=0 for Lu.Source: §2 · p. 4
- Chen and Lu have transition scenarios similar to Lorenz, but their nonlinear forms involve only two parameters and are therefore simpler.The section connects this simplicity to the open question of identifying the algebraically simplest chaotic flow and to a chaotic analogue of Hilbert's 16th problem.Source: §2 · pp. 4–5
3. Recent discussion on equivalence of the Lorenz, Chen, and Lu systems
The section questions whether forward-time analyses and transformations suffice to understand Lorenz, Chen, and Lu systems in backward time. It argues that time direction and parameter signs affect physical interpretation and the existence or analysis of invariant objects, motivating new methods.
3. Recent discussion on equivalence of the Lorenz, Chen, and Lu systems
- The Lorenz system's physically motivated parameters are positive, whereas the transformed systems have negative β for positive b and c; backward-time comparison has no clear physical sense.The paper notes that physical parameter meanings therefore distinguish the systems from Lorenz even where transformations produce a formal reduction.Source: §3 · p. 5
- For non-closed trajectories and forward-invariant sets, studying only forward time may not suffice, even though periodic and homoclinic trajectories can be studied in one time direction.Source: §3 · p. 6
- The cited equivalence discussion leaves solution extension, attractor existence, and invariant sets in backward time or with negative parameters unaddressed.Source: §3 · p. 6
- The paper argues that some Chen and Lu results cannot be derived from Lorenz results because Lorenz repulsor visualization and localization were previously unknown.This counters the cited claim that most Chen-system literature is redundant.Source: §3 · p. 6
- Even when corresponding objects exist in both time directions, their characteristics can differ, and the Lyapunov dimension may be defined for only one time direction.Absolute Lyapunov exponents of a bounded trajectory can also differ between forward and backward time.Source: §3 · p. 7
- The authors identify new methods as necessary to study some properties of time-reversal Lorenz, Chen, and Lu systems.Source: §3 · p. 7
4. Differences and similarities in the analysis of the considered systems
The section contrasts Lorenz-like systems through homoclinic-orbit analysis, divergence-based dimension estimates, stationary-set criteria, and attractor localization. It also identifies how time reversal and attractor type affect their analysis.
4.1. Homoclinic orbits
- For the Lorenz system with σ = 10 and β = 8/3, the stable manifold contains a homoclinic orbit at ρ = 13.9... .Source: §4.1 · p. 7
- Time inversion preserves homoclinic-orbit existence but hinders effective analytical and numerical proof; localization requires many starts near the unstable manifold, with only the homoclinic trajectory returning.Source: §4.1 · p. 9
- Small changes near a saddle can satisfy Shilnikov conditions and yield Shilnikov chaos in the Lorenz, Chen, and Lu systems, although their zero equilibrium is not a saddle-focus.The construction also requires an analog of the Fishing principle and a two-dimensional stable manifold.Source: §4.1 · p. 10
4.2. Divergence and attractor dimension
- When a+b>c, systems (5) and (6), unlike (2) and (3), are not dissipative: volume increases, so bounded sets cannot be positively invariant.Source: §4.2 · p. 11
- If inequality (18) holds throughout U, the Hausdorff dimension of an invariant compact set K ⊂ U is bounded by dimH K < k + s.Source: §4.2 · p. 12
- For classical Chen and Lu systems with d = −1, analogous attractor-dimension expressions are unavailable, and coincidence with the zero-equilibrium value remains conjectural.The conjecture would require new Lyapunov functions and careful consideration.Source: §4.2 · p. 14
5.1. Application of dimension estimates to the problem on stability of stationary sets. Analog of Bendixson criterion
- Under Theorem 5.5's conditions, a continuously differentiable function and nondegenerate matrix S imply that every solution starting in D tends to a stationary set.Source: §5.1 · p. 15
- Under Theorem 5.6's conditions, every bounded forward-time solution of generalized Lorenz system (9) tends to a stationary set.Source: §5.1 · p. 15
6. Conclusion
The conclusion argues that time-inverted reductions of the Chen and Lu systems to Lorenz form create obstacles for analyzing chaotic behavior and attractors. The transformed systems also lack dissipativity, constraining the use of methods that rely on bounded positively invariant sets.
6. Conclusion
- Time-inverted changes of variables reducing Chen and Lu to Lorenz form obstruct attractor visualization, attractor-dimension analysis, and development of methods for studying chaotic behavior.The paper notes that an attractor becomes a repeller under time inversion.Source: §6 · p. 16
- Time inversion also makes effective modern analytical and numerical methods for proving homoclinic trajectories impossible to apply.Source: §6 · p. 16
- The time-inverted Chen and Lu systems have increasing volumes rather than negative divergence, so bounded sets cannot be positively invariant or dissipative in the sense of Levinson.Source: §6 · pp. 16–17
- Time inversion used to reduce Chen and Lu to Lorenz form prevents introducing or estimating Lyapunov dimension.Source: §6 · p. 17