Abstract
Noise modifies the behavior of chaotic systems in both quantitative and qualitative ways. To study these modifications, the present work compares the topological structure of the deterministic Lorenz (1963) attractor with its stochastically perturbed version. The deterministic attractor is well known to be "strange" but it is frozen in time. When driven by multiplicative noise, the Lorenz model's random attractor (LORA) evolves in time. Algebraic topology sheds light on the most striking effects involved in such an evolution. In order to examine the topological structure of the snapshots that approximate LORA, we use branched manifold analysis through homologies-a technique originally introduced to characterize the topological structure of deterministically chaotic flows-which is being extended herein to nonlinear noise-driven systems. The analysis is performed for a fixed realization of the driving noise at different time instants in time. The results suggest that LORA's evolution includes sharp transitions that appear as topological tipping points.
Highlights
AND MOTIVATIONThe use of topological concepts to describe complex physics goes back at least to Lord Kelvin, who proposed that atoms were knotted vortices swirling in the æther, an invisible medium believed at that time to fill the surrounding space[6]
With this criterion of robustness in mind, we require n∗t at each time t to be a value of nat which the H1 generators of the Branched Manifold Analysis through Homologies (BraMAH) cell complex coincide with the most robust 1-holes
We have extended the deterministic concept of branched manifold[2], by defining it locally as an integerdimensional set in phase space that robustly supports the point cloud associated with the system’s invariant measure at each instant t
Summary
AND MOTIVATIONThe use of topological concepts to describe complex physics goes back at least to Lord Kelvin, who proposed that atoms were knotted vortices swirling in the æther, an invisible medium believed at that time to fill the surrounding space[6]. Kelvin’s vision stimulated active research in knot theory[7,8] and eventually led to the awareness that knots and related tangled structures do form in various physical phenomena and can have a pivotal, albeit still poorly understood influence on turbulent fluid dynamics[9,10], quantum field theory[11], and magnetic fields[12,13], to cite but a few areas of physics In his seminal work, Moffatt 14 showed that helicity measures the total knottedness and linkage of a flow and that it is an invariant in ideal, viscosity-lacking fluids, like liquid helium. Lorenz’s famous 1963 paper[21], where he remarks that the flow on the attractor’s “surface” passes “back and forth from one spiral to the other without intersecting itself.” This surface is topologically equivalent to what is called a branched manifold[22]
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