Chaos Theory and Complexity

Dear Colleagues,

Nonlinear dynamical systems mainly involve differential equations and maps. It was Poincaré who first observed that something peculiar happened in the area of homoclinic points that approximated the fixed saddle points of Poincaré maps for differential equations. Lorentz showed that close-but-different orbits depend very closely on initial conditions. Devaney defined chaos.

Autonomous ordinary differential equations can be conservative and even more Hamiltonian and may often show chaos in more than one degree of freedom. Dissipative patterns may lead to chaotic attractors in more than two dimensions. Non-autonomous Hamiltonians and dissipative systems may exhibit chaos at lower dimensions. Accordingly, symplectic maps and dissipative maps may show chaos in one dimension.

Studies on the above-mentioned systems can be performed at low dimensions. For example, for Hamiltonian systems with two degrees of freedom, we can use the Poincaré map. For non-autonomous two-dimensional differential equations, the stroboscopic Poincaré map is another possible tool. The maximum Lyapunov exponent offers numerical proof for the chaoticity or lack of it in a system. Another method of numerical investigation is the bifurcation diagram. We can also find the fractal or the Hausdorff dimension of a chaotic attractor or the Kaplan–Yorke dimension.

Theoretical methods can vary from abstract topological spaces to applications of theorems, such as Melnikov’s homoclinic and subharmonic theory or Poincaré’s nonintegrability theorem, to specific problems. The above theoretical and numerical methods can be applied in fields including mechanical systems, nonlinear electric circuits, electronics, biology, economics, and medicine.

Complexity is a vast area of investigation and concerns the above-mentioned nonlinear dynamical systems and partial differential equations, networks (graphs), fractals in the solid state, nanoscience, etc. Partial differential equations are of great importance in theories regarding the existence of solutions and integrability, but little research has been carried out on chaos and complexity. There are many applications, such as in diffusion or plasma physics. Network theory is a much more recent development, and, if we wish to apply networks, we should work numerically. These networks are applied in fields including economic networks, traffic, and epidemiology.

Dr. Efthymia Meletlidou
Guest Editor

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• chaos
• dynamical system
• complexity
• nonlinear dynamics