This work sits at the intersection of numerical analysis of dynamical systems and chaos theory. Surrounding research includes studies on orbital dynamics, hydrodynamic models, and entropy characteristics, but this article stands out by focusing on the fundamental problem of distinguishing sustained chaos from transient chaotic sets in classical systems. Neighboring works often assume that a numerically detected chaotic attractor truly exists; here, it is shown that this assumption can be wrong. This is particularly important for models where transient times can be enormous, casting doubt on the reliability of many computational results.
The analysis used the classical Lorenz, Rössler, and Vallis systems. In the Lorenz system with parameters r=24, σ=10, b=8/3, a hidden transient chaotic set was visualized; its trajectories can behave chaotically for a long time before converging to a stable equilibrium. For the Rössler system, stabilization of an unstable periodic orbit via Pyragas control was applied, allowing comparison of the behavior of stabilized and non-stabilized trajectories when computing finite-time Lyapunov exponents. For the Vallis system, an analytical approach based on Lyapunov functions and the Douady–Oesterlé method was used, yielding an estimate of the Lyapunov dimension of the global attractor without numerical simulation.
In the Lorenz system at r=24, a hidden transient chaotic set was found: the duration of chaotic behavior depends on initial conditions and can exceed $10^7$ time units for the point (0,-0.5,0.5), whereas for the point (20,20,20) chaos lasts only about 1.8·$10^4$. This shows that numerical detection of an attractor can be an illusion. In the Rössler system, it was shown that when computing the largest finite-time Lyapunov exponent along an unstable periodic orbit, the results for stabilized and non-stabilized trajectories agree to the 5th decimal place only on the interval [0,30.4], after which they diverge. For the Vallis system, an analytical estimate of the Lyapunov dimension of the global attractor was obtained: dim_L $A_V$ ≤ 3 - 2(C+2)/(C+1+√((C-1)^2+25B/4)), which is consistent with local dimensions at equilibria.
This work differs from neighboring studies by questioning the very possibility of reliably determining chaoticity numerically in some systems. While many works use long computations to characterize attractors, here it is shown that even with very long integration times, one can misclassify a set as an attractor. This underscores the need to combine numerical methods with analytical approaches, such as Lyapunov functions, for reliable localization of attractors. Such an approach is especially valuable for systems where transient chaos can last unpredictably long.
The topic could develop toward more reliable criteria for distinguishing attractors from transient chaotic sets based on finite-time characteristics. Algorithms might be created that automatically determine whether observed chaos is sustained or transient, using grid analysis of initial conditions. Also promising is the application of analytical estimates of Lyapunov dimension to a wider class of systems, including climate and turbulence models, where numerical computations are particularly sensitive to errors.
The results will affect computational fluid dynamics, climate theory, and nonlinear dynamics, where the reliability of numerical predictions is critical. They will also find application in control systems engineering, where it is necessary to distinguish stable regimes from transient ones.
The next step could be a systematic study of hidden transient chaotic sets in other well-known systems, such as the Rabinovich system or the Glukhovsky–Dolzhansky system. It is also important to develop methods for estimating the lifetime of transient chaos based on analytical properties of the system.
The article is directly related to the open problem of the existence of hidden attractors in the Lorenz system, which is a generalization of the second part of Hilbert's 16th problem to chaotic systems. This problem remains unsolved, and this work contributes to understanding why it is so difficult to solve numerically.
In the Lorenz system at r=24, chaotic behavior can last more than 10 million time units before the trajectory converges to a stable equilibrium. That is like watching a storm in a glass of water for a year and then having it suddenly calm down.