About the oscillatory possibilities of the dynamical systems

Abstract

This paper attempts to make feasible the evolutionary emergence of novelty in a supposedly deterministic world which behavior is associated with those of the mathematical dynamical systems. The work was motivated by the observation of complex oscillatory behaviors in a family of physical devices, for which there is no known explanation in the mainstream of nonlinear dynamics. The paper begins by describing a nonlinear mechanism of oscillatory mode mixing explaining such behaviors and establishes a generic dynamical scenario with extraordinary oscillatory possibilities, including expansive growing scalability. The relation of the scenario to the oscillatory behaviors of turbulent fluids and living brains is discussed. Finally, by considering the scenario as a dynamic substrate underlying generic aspects of both the functioning and the genesis of complexity in a supposedly deterministic world, a theoretical framework covering the evolutionary development of structural transformations in the time evolution of that world is built up.

Figures (12)

• Figure 1: Phase space representation of numerical results for an $N=3$ system illustrating the nonlinear mixing of two oscillation modes emerged from a saddle-node pair of fixed points. The Hopf bifurcations take place at $\mu_{C}$=12.96 on the saddle $S_{1}$ and at 13.10 on the node $S_{0}$. The represented stable orbit has grown with $\mu_{C}$ incorporating helical turns around the unstable manifold (in grey) of the saddle cycle (broken), without undergoing any bifurcation and remaining strictly periodic. The inset shows the time evolution of one of the variables ($\psi$) for three control parameter values: from just after the bifurcation up to the represented attractor.
• Figure 2: Numerical results for $N=4$ illustrating how two oscillation modes emerged in successive Hopf bifurcations of the same fixed point mix without a torus bifurcation. The Hopf bifurcations take place at $\mu_{C}$=8.2 and 9.1 with angular frequencies of 1.41 and 25, respectively. The stable orbit (black) has grown with $\mu_{C}$ incorporating helical turns around the three-dimensional unstable manifold (in grey) of the saddle cycle (broken white), without undergoing any bifurcation up to $\mu_{C}$=12, where a period doubling occurs. Label a denotes where a second helical structure will appear. The unstable manifold is represented through a few of the trajectories to facilitate its visualization. The saddle cycle does a subcritical torus bifurcation at $\mu_{C}$=10.3 with a secondary frequency of 1.38 and it becomes stable within a saddle torus. The inset shows the time evolution of one of the variables ($\psi$) for the stable orbit at different control parameter values.
• Figure 3: Numerical results showing the nonlinear mixing of 5 oscillation modes in the time evolution of an $N=6$ system for the given value of the control parameter $\mu_{C}$. The system of equations (2)-(4) has been designed (see Subsection 3.2) by imposing Hopf bifurcations of angular frequencies 0.04, 0.25, 2, 20, and 125, alternatively in a saddle-node pair of fixed points and according to the values of $p$ equal to -6, 15.6, -7.4, 16.2, and -6.9, and by choosing $c_{1}= 250$. The nonlinear function is $g(\psi)=(1.25-1.06\cos\psi)/(1.68-\cos\psi)$, with which the Hopf bifurcations on the involved saddle-node pair of fixed points happen at $\mu_{C}$= 38.5, 53.5, 40.0, 54.8, and 39.4, respectively. Numeric labels denote the different modes ordered from lower to higher frequencies. Odd (even) numbers correspond to the node (saddle) point. More than 200 previous cycles have been discarded in the represented signal to assure its approach to asymptotic behavior. The orbit periodicity has not been strictly verified but the successive cycles show extremely similar structure. The oscillation modes 2 and 4 appear on the time evolution even if the corresponding Hopf bifurcations will occur at higher $\mu_{C}$ values, suggesting the subcriticality of such bifurcations.
• Figure 4: The orbit of Fig. \ref{['Fig2']} projected in different planes of the six-dimensional phase space. The fixed points are located on the $z_{N}$ axis ($S_{0}$ and $S_{1}$ are at $d_{6}z_{6}$ equal to 29.7 and 31.9) and then the two points appear superposed at the origin of the planes perpendicular to this axis. Numeric labels denote the different oscillatory modes.
• Figure 5: Arrays of fixed points within $m$-dimensional linear subspaces of the phase space, for $N$-dimensional systems having efficient nonlinearities in $m$ linearly-independent components of the vector field. Numeric labels denote unstable manifold dimensions of the equilibria by supposing the dimensions outside the $m$-dimensional subspaces to be attractive for physical reasons. Regular arrays are drawn for simplicity but both the separation distance and position alignment of equilibria will be surely irregular within the linear subspaces. Only one attraction basin is represented for $m=2$ and only one of the basin corners is represented for $m=3$. In general, the arrays would extend at the other side of the separatrix with additional attractors.