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Overview of dynamical systems and chaos
Published in Marcio Eisencraft, Romis Attux, Ricardo Suyama, Chaotic Signals in Digital Communications, 2018
In time-continuous systems, trajectories can lie on (hyper)surfaces. For instance, in a three-dimensional system, a (two-dimensional) surface of a torus can be an invariant set. A closed trajectory lying on this toroidal surface corresponds to a periodic motion; a trajectory that runs on such a surface and never closes itself implies a quasiperiodic motion. To clarify this issue, consider a plane perpendicularly crossing this two-dimensional torus, such that the intersection between them is a circumference. Such a plane can be viewed as a Poincaré section. Suppose that consecutive intersections of a trajectory with this vertical plane are represented by the Poincaré map x1(t + 1) = x1(t), x2(t + 1) = x2(t) + 2πl. Here, x1(t) is the radial position and x2(t) is the angular position of an intersection in the time step t. The initial condition is given by (x1(0), x2(0)). Because the trajectories lie on the toroidal surface, their radial positions are kept constant (these intersections occur on the circumference of radius x1(0)). Notice that if l is a rational number, that is, l = l1/l2(l1, l2 ∈ ℤ+), then the orbit of this map returns to its initial angular position x2(0) after l2 time steps. Consequently, the trajectory of the continuous-time system closes itself and it corresponds to a periodic motion. However, if l is an irrational number (l cannot be written as l = l1/l2), then an orbit of the difference equations never returns to the initial state. Therefore, the trajectory of the continuous-time system never closes itself; in fact, it densely fills the toroidal surface, which is interpreted as a quasiperiodic behavior. In this case, any state is never exactly repeated.
The chaotic, supernonlinear, periodic, quasiperiodic wave solutions and solitons with cascaded system
Published in Waves in Random and Complex Media, 2021
Nauman Raza, Adil Jhangeer, Saima Arshed, Mustafa Inc
In Figure 4, 2D and 3D phase portraits and the time series graph and Poincaré section are presented for and . Chaotic behavior can be seen for these values of physical parameters. While from Figure 5, it is observed that there is no pattern in Poincaré section which leads towards the observation that chaotic behavior is present for these particular values of physical parameters. In Figure 8, by altering the number of perturbed term equal to zero and keeping other parameters same as taken in Figure 4, supernonlinear and nonlinear periodic waves are determined escorted by Runge–Kutta method. In Figure 6, we illustrated the 2D and 3D phase portraits and the time series graph for by assuming an other parameter as it is taken in Figure 4. This is observed that the dynamical system (67) carries quasiperiodic motion. Poincaré section for these parameters are given in Figure 7 which takes a particular form for these values that nullify the existence of chaotic behavior (Figure 8). In Figure 9, after neglecting the perturbed term of the system (67) and taking other parameters same as taken in Figure 5, with the help of Runge–Kutta method supernonlinear and nonlinear periodic waves are obtained.