Chaos in 3D and 4D Thermodynamic Models

Recently, Aydiner considered dark matter (DM) and dark energy (DE) as two open, non-equilibrium thermodynamic systems, which have heat changes and particle number changes but have no volume changes. These systems are described by nonlinear coupled equations for the description of mutual and self-interactions and satisfy the energy conservation of thermodynamics. Based on this idea, two three-dimensional (3D) models and a four-dimensional (4D) model are produced. Due to the conservation of the energy–momentum tensor of the sum of the DM and DE energy densities, the continuity equations of both energy densities are also included together in these 3D and 4D thermodynamic models. For the parameters satisfying some conditions, one of the 3D models has two marginal stable non-hyperbolic equilibrium points with a negative real root and a pair of conjugate purely imaginary roots. The marginal stability is highly sensitive to nonlinear terms and parameter noise. Another of the 3D models has unstable saddle-focus equilibrium points, which have a negative real root corresponding to a 1D stable manifold and two conjugate complex roots with positive real parts corresponding to a 2D manifold of unstable spiral. At these equilibria, no energy exchange occurs between the two energy densities, and both energy components reach equilibrium. When some perturbations from the nonlinear terms or parameter noise are given, the DM and DE energy densities are far from equilibrium and continue to exchange each other until they reach equilibrium. The energy exchanges between them may exhibit chaotic behavior like chaotic attractors. However, hyperchaos is not easily found. The 4D model also has unstable saddle-focus equilibrium points and can allow for the onset of chaotic attractors and hyperchaos. In fact, the chaotic dynamics of the 3D and 4D models are caused because of the coupled interactions of particle and thermodynamic systems between DM and DE. Under both the self-interactions and the mutual interactions, the energy exchanges are far from and close to the equilibrium. These interactions cause the energy exchanges to become random, irregular and unpredictable.