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#Space billiards series
Also, when one has only a time series of observations, the algorithm proposed by Wolf et al. In such cases, cloned dynamics method 22 can be employed. In certain situations, the linearized version of the state equations is unavailable due to presence of discontinuities. From this, and a generic initial perturbation, we appropriately estimate the most expanding direction of deviations. The standard method for calculating \(\lambda\) is the so-called tangent space method that consists in the integration of the system’s equation of motion with its linearized versions considered in successive time steps 21. When \(\lambda > 0\), there exists at least one direction in phase space in which the dynamics is unstable implying chaotic behavior. This quantity is related to the linear stability of a trajectory. The Lyapunov exponent (LE), historically denoted by \(\lambda\), is an objective measurement of chaos. In addition to the ergodic properties, BSB is also a mixing system with Kolmogorov property 19 and Bernoulli property 20, which means that regular and chaotic dynamics coexist.
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The BSB and SSB are examples, therefore, of ergodic dynamics 19, which assert that, under certain conditions, the time average of a function along the trajectories exists almost everywhere and is related to the space average. This means, then, the dynamics is sensitive to the initial conditions, is non-periodic, and thereby the particle accesses all the regions of the stadium. 1a, or in Sinai stadium billiards 18 (SSB), the sequence of straight flights and angles is a non-trivial time series. However, when the boundary is shaped as in Bunimovich stadium billiards 17 (BSB), as illustrated by Fig. Consequently, the particle carries out periodic orbits, visiting only a part of the internal area of the stadium. In this case, the sequence of straight flights and reflection angles produce a regular time series. For example, in circular stadium billiards 16 (CSB), the particle’s evolution is insensitive to the initial conditions. The motion in billiards consists of a sequence of straight flights interrupted by specular reflections. This system has two axes of symmetry which can be also used to increase the number of trajectories in it. The dynamics of billiards is probably the simplest system in which chaos emerges 10, 15. The hallmark of chaos is the super-sensitivity of a nonlinear system to its initial conditions, which makes classical dynamics appear unpredictable over long periods of time 3. It provides a collection of concepts and methods to analyze a novel behavior emerging in a wide range of disciplines 1. The influence of chaos theory extends beyond physics and mathematics to include biology, chemistry, communications, engineering, cryptography, and robotics 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. Lorenz’s work in meteorology 4 is the modern foundation stone of chaos theory. Chaos is a theory that has ramifications in all of science 1, 2, 3.