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For bimodal maps the Feigenbaum’s renormalization group (RG) equation is generalized to a system of two independent RG equations, each corresponding to a critical point. A series of numerical solutions to systems of the RG equations are obtained, which explain the metric universalities of period-tripling and period-quadrupling bifurcations. For a special type of period-p-tupling bifurcations, it is found that the “collision” of the two critical points of bimodal maps leads to a quadratic relation between the two universal scaling factors. [Copyright &y& Elsevier]

For bimodal maps the concept of an equal topological entropy class (ETEC) is established by the dual star products. All the infinitely many ETEC plateaus and single points are harmonically organized in the kneading parameter plane, they construct a multifractal devil’s carpet, which possesses a perfect subregion similarity and a dual central symmetry. The entropy devil’s carpet reveals the complexity of global dynamical behavior in the whole parameter plane of bimodal systems. [Copyright &y& Elsevier]

In one-dimensional chaotic dynamics, a global multifractal relation between topological entropies and fractal dimensions of arbitrary period-p-tupling attractors is analyzed on all critical (accumulation) points of transitions to chaos, where the Lyapunov characteristic exponent is zero. The global metric regularity of topological entropies versus fractal dimensions is well characterized by the self-similarity. By the fractal interpolation based on the iterated function system, the fractal dimensions of the curves of topological entropies versus capacity dimensions and versus information dimensions are both found to be 1.82. [Copyright &y& Elsevier]