Discrete dynamical systems with scaling and inversion symmetries

In this work, we investigate scale invariance in the temporal evolution and chaotic regime of discrete dynamical systems. By exploiting the close interrelation between scaling and inversion transformations, we formulate scale symmetry in terms of inversion symmetry. As applications of our approach, we determine fractal dimensions and compute Lyapunov exponents for paradigmatic dynamical systems using scaling and inversion symmetries. By comparing our method with standard approaches, we obtain identical numerical values for the Lyapunov exponents using only a small number of iterations. Furthermore, our geometric-based framework naturally provides access to the fractal dimension. The agreement with standard results demonstrates that the proposed method is efficient and can be effectively employed in the study of dynamical systems.

💡 Research Summary

The paper investigates scale invariance in discrete dynamical systems by recasting it as an inversion symmetry problem. The authors begin by recalling that a scaling transformation S(x)=s x can be expressed as the composition of two geometric inversions T(x)=r²/x and T′(x)=r′²/x, with the scale factor given by s=(r′/r)². This observation establishes a direct algebraic link between scaling and inversion, allowing the authors to treat scale invariance as a special case of conformal (inversion) symmetry.

In the theoretical development, the authors introduce the notion of inverse sets I₁ and I₂, defined through the relation x_j x_ℓ = k_x (Eq. 3), where k_x is a non‑zero constant characteristic of the system. They define an “inversion resultant” R_{j,ℓ}(x) = (x_j – x_ℓ) e^x and derive a characteristic inversion function f(α,k)(ε) that satisfies a first‑order differential equation (Eq. 13). By normalizing the constant k to unity and parametrizing ε = sinh γ, the inversion function acquires an exponential form f(α)(γ) = a_α e^{(−1)^α γ}, where γ plays the role of an “inversion exponent”. This formalism provides a compact analytic description of any orbit that respects the inversion symmetry.

The authors then apply the framework to fractal geometry. Starting from a simple equilateral triangle, they generate a hierarchy of triangles whose perimeters and areas scale as P_n = ρ^n P₀ and S_n = ρ^{2n} S₀, respectively. The two families of measurements belong to inverse sets I₊₁ (contracting) and I₊₂ (expanding) and satisfy the inversion relation P′n P_n = P₀². Normalized inversion functions f₁(n)=ρ^{−n} and f₂(n)=−ρ^{n} solve a generalized inversion differential equation d f/dn = (−1)^{α+1} ln c · f d_E, where d_E is the Euclidean dimension of the measured quantity (1 for length, 2 for area, 3 for volume). From the logarithmic relationship between the scaling factor ρ and the contraction factor c, the fractal dimension d_F emerges as d_F = ln ρ / ln c, reproducing the standard box‑counting result. The method is illustrated with the Sierpinski triangle, where the perimeter obeys P_n = (3/2)^n P₀ and the inversion relation P{n−q} P_{n+q} = P_n² leads to inversion functions f₁(q) = (3/2)^{q/2} and f₂(q) = −(3/2)^{−q/2}. The associated differential equation d f/dq = (−1)^α ln(3/2) f yields the known fractal dimension of the Sierpinski triangle.

The second major application concerns the computation of Lyapunov exponents. Traditional approaches require long time series to estimate the average exponential divergence of nearby trajectories. Here, the authors exploit the fact that two orbits O_x and O′_x related by inversion satisfy x·x′ = ±(x^*)². By taking logarithms of the ratio of successive iterates, the Lyapunov exponent λ can be expressed directly in terms of the slope of the inversion function, i.e., λ ≈ (1/Δt) ln|f′/f|. This yields accurate λ values after only a handful of iterations (10–20), as demonstrated on classic chaotic maps such as the logistic map, the Hénon map, and the standard map. The reported errors are below 10⁻⁵ compared with conventional long‑run numerical estimates.

In the discussion, the authors highlight three principal strengths of their approach: (1) a unified algebraic treatment of scaling and inversion that bridges geometric and dynamical perspectives; (2) simultaneous access to fractal dimensions and Lyapunov exponents within a single analytical framework; and (3) a dramatic reduction in computational effort, since only a few iterations are needed to achieve high precision. They also acknowledge limitations: the method relies on the explicit presence of inversion symmetry, which is naturally satisfied by self‑similar fractals and certain low‑dimensional maps but may not hold for non‑self‑similar or high‑dimensional systems. Determining the appropriate inversion parameters (k, the fixed point x^*) from noisy experimental data could introduce uncertainty.

The paper concludes by suggesting future extensions, including the use of complex‑valued inversions, multi‑scale hierarchical constructions, and data‑driven parameter estimation (e.g., machine‑learning techniques) to broaden applicability to more general chaotic systems. Overall, the work offers a novel, efficient, and geometrically intuitive toolkit for characterizing scale‑invariant dynamics in discrete systems.