Fig. 1

Visualization of the Rössler attractor, a continuous-time dynamical system that exhibits chaotic dynamics with fractal properties. The system is composed of three nonlinear differential equations and thus has three variables. a Temporal fluctuations of the x variable. Although each variable oscillates within a fixed range of values, its amplitude is highly variable. b Three-dimensional trajectory of the Rössler attractor visualized using the values of all three variables. c Three-dimensional plot of the Rössler attractor reconstructed using delay-time embedding (\(\tau =30\,\mathrm{ a}.\mathrm{u}.\)) of a single variable (x). Note that the complexity of the system is well preserved compared to b, in which all three variables are used. d Recurrence plot of the reconstructed attractor using delay-time embedding (\(m=3, \tau =30\,\mathrm{ a}.\mathrm{u}.\), r = 1.0) consisting of diagonal lines with different lengths, indicating that the trajectories pass neighborhoods of past trajectories with limited durations. This means that the reconstructed attractor exhibits a fractal structure or self-similarity