Scaling invariance: a bridge between geometry, dynamics and criticality

Scale invariance is a central organising principle in physics, underlying phenomena from critical behaviour in statistical mechanics to transport and chaos in nonlinear dynamics. We present a unified exploration of scaling concepts, emphasising how invariance under rescaling emerges across systems of increasing dynamical complexity. Beginning with elementary models governed by a single control parameter, we show how power-law behaviour arises in the absence of characteristic scales. We then examine nonlinear systems with local bifurcations, where scaling variables control relaxation toward stationary states and reveal critical exponents, crossover phenomena, and critical slowing down. Finally, we analyse continuous transitions in chaotic systems using concepts such as order parameters and symmetry breaking. These examples demonstrate that scaling provides a unifying framework linking deterministic dynamics and nonequilibrium statistical physics across diverse physical contexts.