Dynamics of Complex Systems and Their Associated Attractors in a Multifractal Paradigm of Motion

In this paper we analyze complex systems dynamics using a multifractal framework derived from Scale Relativity Theory (SRT). By extending classical differential geometry to accommodate non-differentiable, scale-dependent behaviors, we formulate Schrödinger-type equations that describe multifractal geodesics. These equations reveal deep analogies between quantum mechanics and macroscopic complex dynamics. A key feature of this approach is the identification of hidden symmetries governed by multifractal analogs of classical groups, particularly the SL(2ℝ) group. These symmetries help explain universal dynamic behaviors such as double period dynamics, damped dynamics, modulated dynamics, or chaotic dynamics. The resulting framework offers a unified geometric and algebraic perspective on the emergence of order within complex systems, highlighting the fundamental role of fractality and scale covariance in nature.