Mathematical problems of nonlinear dynamics: A tutorial Leonid Shilnikov Abstract We review the theory of nonlinear systems, especially that of strange attractors, and give its perspectives. Of a special attention are the recent results concerning hyperbolic attractors and features of high-dimensional systems in the Newhouse regions. We present an example of a “wild” strange attractor of the topological dimension three. 1 Content 1. Introduction 2. Basic notions of the theory of dynamical systems 3. The Andronov-Pontryagin theorem. Morse-Smale systems 4. Poincar´ehomoclinic structures 5. Structurally stable systems 6. The bifurcation theory. Appearance of hyperbolic attractors 7. Structurally unstable systems. Wild hyperbolic sets. Newhouse regions 8. Lorenz attractor 9. Quasiattractors. Non-transverse homoclinic curves 10. Example of wild strange attractor 2 1 Introduction The early 60th is the beginning of the intensive development of the theory of high-dimensional dynamical systems. Within a short period of time Smale [66, 67] had established the basics of the theory of structurally unstable systems with the complex behaviour of trajectories, the theory which we now know as the hyperbolic theory. In essence, a new mathematical discipline with its own terminology, notions etc, has been created, which at the same time interacts actively with other mathematical disciplines. Here we must emphasize the role of the qualitative theory of differential equation (QTODEs). In fact, this theory provides a foundation for investigating many problems of natural sciences and engineering which have a nonlinear dynamics origin. On the other hand the qualitative theory of differential equations itself takes new ideas from nonlinear dynamics. The usefulness and the necessity of such a synthesis were clear for such scientists with the broad vision on science as Poincar´eand Andronov. All this makes the QTODEs especially attractive and practical. Its achievements have led to one of the brightest scientific discoveries of the XX century — dy- namical chaos. Since the moment of the discovery of dynamical chaos along with such customary dynamical regimes as stationary states, self-oscillations and modulations, chaotic oscillations entered the science. If the mathemati- cal images of the formers are equilibrium states, periodic orbits and tori with quasiperiodic trajectories, then the adequate image of dynamical chaos is a strange attractor, i.e., an attractive limiting set with the unstable behaviour of its trajectories. Those attractors that persist this property under small smooth perturbations will interest us in this review. Namely, such attractors have been predicted by the hyperbolic theory of high-dimensional dynamical systems. However, the role and the significance of strange attractors were not accepted by researchers of certain scientific directions, in particular of turbulence, for a sufficiently long time. There were a few reasons for that. The hyperbolic theory had examples of strange attractors but their structure were so topologically 3 complex that did not allow one to imagine rather simple scenarios of their origination which is very important for nonlinear dynamics, which deals with models described by differential equations. On the other hand, those “strange attractors” observed in concrete models were not hyperbolic attractors in the strict meaning of these words. Most of them, possessing all the properties of a “qenuine” strange attractor, had stable periodic orbits. This gave a chance to argue that the observable chaotic behaviour is intermittent. Here, we have to bear in mind that when speaking about dynamical systems we are interested not in the character of a solution over some bounded period of time but in the information on its limiting behaviour when time increases to infinity. Note also strange attractors that have hyperbolic subsets co-existing with stable long periodic orbits of very narrow and tortious attraction basin, the so-called quasi- 1attractors [10]. The breakthrough came in the mid 70-ths with the appearance of a “simple” low-dimensional model x˙ = σ(x y), y˙ = rx− y− xz, z˙ = bz−+ xy− − in which Lorenz had discovered numerically in 1962 a chaotic behaviour in the trajectories. A detailed analysis carried out by mathematicians revealed the existence of a strange attractor, not hyperbolic but non-structurally stable. Nevertheless, the main feature — the instability of the behaviour of trajectories under small smooth perturbations of the system — of this attractor persists. Such attractors, which contain a single equilibrium state of the saddle type, will be henceforth be called Lorenz(ian) attractors. The second remarkable fact related to these attractors is that the Lorenz attractor may be generated on the route of a finite number of rather simple observable bifurcations from systems with trivial dynamics. Since that time the phenomenon of dynamical chaos was “almost legislated”. In this breakthrough, the fact that the Lorenz model came from hydrodynamics played not the least but a primary role. The topological dimension of the Lorenz attractor, regardless of the dimen- 1stochastic 4 sion of the associated concrete system, is always two and its fractal dimension is less than three. At the same time, researchers who deal with extended sys- tems often observe chaotic regimes of presumably much higher dimensions. It is customary then to say that hyperchaos occurs. But which attractors describe hyperchaos? Are they strange attractors or quasiattractors? In principal, the hyperbolic theory predicts the possibility of the existence of strange attractor of any finite dimensions. The tragedy is that nobody has observed known hy- perbolic attractors in nonlinear dynamics since the moment of their invention. There has been some progress recently: the author and Turaev [79] have proved that a number of hyperbolic attractors (structurally like the Smale-Williams solenoids and the Anosov tori) may be obtained through one global bifurcation of the disappearance of a stable periodic orbit or of an invariant torus with a quasi-periodic trajectory on it. They have also discovered a principally new type of strange attractor — the so-called wild strange attractors. Their distinc- tion from the known attractors is that they contain an equilibrium state of the saddle-focus type as well as saddle periodic trajectories of various types, namely, dimensions of invariant manifolds of the co-existing trajectories may be equal both two and three. Moreover, the region of the existence of such an attractor is a region of everywhere dense structural instability due to homoclinic tangencies. Thus, the above phenomenon poses principally new problems for ergodic theory. Due to the co-existence of the trajectories of various types the wild attractors, as well as Lorenz-like attractors, are the pseudo-hyperbolic attractors. Since the notion of pseidohyperbolicity, which plays a dominating role in the theory of structurally unstable strange attractors, will be used below. So let us stop here and discuss it in detail. Consider a smooth n-dimensional dynamical system x˙ = X(x), in a bounded region D which satisfies the following conditions: 1. On its boundary ∂D the vector flow goes inward D. This implies that for any point x ∂D either an entire trajectory or a semitrajectory is defined ∈ that passes through the point x. 5 2. A pseudo-hyperbolicity takes place in D. This implies that at each point x D the tangent space, invariant with respect to the associated lin- ∈ earized flow, may be decomposed as a direct sum of subspaces N1 and N2, depending continuously on the point x so that the maximal Lyapunov ex- ponent, corresponding to N1, is strongly less than any Lyapunov exponent corresponding to N2. In other words, the associated variational equation can be represented in the form ξ˙ = A1(t)ξ, η˙ = A2(t)η, where the contraction in ξ is stronger than the contraction in η. 3. The linearized semiflow is volume-expanding V const eσtV , σ> 0. t ≥ 0 Note that the property of the pseudo-hyperbolicity persists under small smooth perturbations, as does the property of the exponential expansion of volumes in N2. Due to the above requirements there will exist at least one strange attractor in the region D. Note that this suggested criterion on the existence of a pseudo- hyperbolic attractor in D is formulated relatively simply, but, similarly the principle of the contraction mappings, its real verification in concrete systems will not be trivial. Let us return to the problem of quasi-attractors. In both situation, in the case of quasi-attractors and in the case of wild strange attractors, the reason of complexity is the presence of structurally unstable Poincar´ehomoclinic curves, i.e., bi-asymptotic trajectories to a saddle periodic orbit, along which its stable and unstable manifolds have a non-transverse contact. This type of homo- clinic trajectories is also responsible for the existence in the space of dynamical systems of regions of everywhere dense structural instability — the so-called Newhouse regions in which systems with homoclinic tangencies are dense. Un- der certain conditions systems with infinitely many stable periodic orbits are also dense in the Newhouse regions. At least this is always true for three- dimensional systems with negative divergence. The peculiarity of such a set of 6 stable periodic orbits is that it cannot be separated in a quasiattractor from the co-existing hyperbolic subset to which these periodic orbits accumulate. In three-dimensional