Newhouse phenomenon
The Newhouse phenomenon is the existence of open regions in the parameter space of smooth dynamical systems in which the system possesses infinitely many coexisting periodic attractors. Discovered by Sheldon Newhouse in the 1970s, the phenomenon shattered the prevailing assumption that chaotic systems could be classified into a finite number of structurally stable types. It showed that beyond the well-understood world of hyperbolic systems lies a wilderness of non-hyperbolic dynamics that is not merely more complex but fundamentally different in kind.
The phenomenon arises in the context of homoclinic bifurcations — the creation and destruction of transverse homoclinic points as a parameter is varied. Newhouse proved that near certain homoclinic tangencies (non-transverse intersections of stable and unstable manifolds), there exist open intervals of parameter values for which the corresponding diffeomorphism has infinitely many periodic sinks. These sinks are not isolated anomalies; they are dense in the parameter interval, and their basins of attraction are interwoven in a complex, fractal pattern.
The Mechanism: Homoclinic Tangencies and Persistent Tangencies
The key to the Newhouse phenomenon is the concept of a persistent homoclinic tangency. In a hyperbolic system, stable and unstable manifolds intersect transversely, and this transversality persists under small perturbations. But when a parameter is varied, a transverse intersection can become tangential at a critical parameter value, and then disappear or reappear on the other side. Newhouse showed that if the tangency is sufficiently degenerate — specifically, if the stable and unstable manifolds have different curvatures at the point of tangency — then there exist open sets of parameters near the tangency for which the tangency persists.
This persistence is counterintuitive. One would expect that a tangency, being a codimension-one phenomenon, would be destroyed by a generic perturbation. And indeed, for a fixed parameter value, a tangency can be destroyed by a perturbation of the map. But Newhouse's theorem is about parameter intervals: there exist open intervals of parameters for which every map in the interval has a homoclinic tangency. The tangency is not persistent for a single map; it is persistent for a set of maps that is open in parameter space.
The mechanism involves the creation of new periodic orbits with each perturbation. Each time a homoclinic tangency is created, it produces a cascade of period-doubling bifurcations and the birth of new periodic sinks. Because the tangency is persistent in parameter space, this process repeats infinitely, producing infinitely many periodic attractors. The attractors are not all present at a single parameter value; rather, they accumulate at certain parameter values, creating a dense set of parameters at which the system has infinitely many attractors.
The Hénon Map and the Paradigm Shift
The Newhouse phenomenon is not merely a theoretical curiosity. It appears in concrete systems. The Hénon map, the paradigmatic example of a two-dimensional chaotic map, exhibits the Newhouse phenomenon for certain parameter values. Numerical studies have confirmed the existence of parameter regions with many coexisting attractors, and the bifurcation diagrams of the Hénon map show the complex intertwining of periodic windows that is the signature of the phenomenon.
The significance for the classification of dynamical systems is profound. The structural stability program, championed by Stephen Smale, aimed to classify dynamical systems into a finite number of equivalence classes, each represented by a structurally stable system. The Newhouse phenomenon shows that this program cannot succeed for generic smooth systems. There are regions of parameter space that are not structurally stable, that contain no structurally stable systems, and that are dense in the space of all systems. The classification problem is not merely difficult; it is impossible in the form originally envisioned.
This does not mean that dynamical systems theory is hopeless. It means that the goal must be reframed. Instead of classifying all systems into a finite number of types, the theory must aim to understand the mechanisms that produce complexity, the statistical properties of typical systems, and the ways in which non-hyperbolic dynamics can be understood through partial hyperbolicity, Young towers, and other constructions. The Newhouse phenomenon is not a failure of dynamical systems theory; it is a discovery that forces the theory to grow.
The Persistent Attractor Problem
A deeper question raised by the Newhouse phenomenon is the persistent attractor problem: are there open regions of parameter space in which generic systems have no persistent attractors at all? That is, are there regions where the attractors are all transient, where the system's long-term behavior is not captured by any finite set of attractors?
This question is related to the broader problem of strange nonchaotic attractors and rank-one attractors, which are attractors that are not hyperbolic but still have well-defined statistical properties. The work of Wang and Young on rank-one attractors, and the theory of heteroclinic cycles and Arnold diffusion, are part of the same effort to understand dynamics beyond hyperbolicity.
The Newhouse phenomenon also has implications for the physical interpretation of dynamical systems. In a system with infinitely many attractors, the long-term behavior depends sensitively on the initial condition not just in the exponential sense of chaos but in the topological sense of which basin the initial condition falls into. Two nearby initial conditions may converge to different attractors, and the boundary between basins may be fractal. This is a stronger form of unpredictability than chaos alone: not only is the trajectory unpredictable, but the attractor itself is unpredictable.
The Newhouse phenomenon is the proof that chaos has depths beyond the horseshoe. The hyperbolic theory gave us the grammar of chaos; the Newhouse phenomenon gives us the poetry — messy, irreducible, and infinitely richer than any finite classification could capture.