Volume 30, Number 2
Special Issue: In Honor of the 90th Anniversary of L.P. Shilnikov: Part II (Guest Editors: Sergey Gonchenko, Andrey Shilnikov, Dmitry Turaev, and Alexey Kazakov)

We review the works initiated and developed by L.P. Shilnikov on homoclinic chaos, highlighting his fundamental contributions to Poincar´e homoclinics to periodic orbits and invariant tori. Additionally, we discuss his related findings in non-autonomous and infinitedimensional systems. This survey continues our earlier review [1], where we examined Shilnikov’s groundbreaking results on bifurcations of homoclinic orbits — his extension of the classical work by A.A. Andronov and E.A. Leontovich from planar to multidimensional autonomous systems, as well as his pioneering discoveries on saddle-focus loops and spiral chaos.

We describe a $C^1$-open set of systems of differential equations in $R^n$, for any $n\geqslant 4$, where every system has a chain-transitive chaotic attractor which contains a saddle-focus equilibrium with a two-dimensional unstable manifold. The attractor also includes a wild hyperbolic set and a heterodimensional cycle involving hyperbolic sets with different numbers of positive Lyapunov exponents.

Recent PDE studies address global boundedness versus finite-time blow-up in equations like the quadratic parabolic heat equation versus the nonconservative quadratic Schrödinger equation. The two equations are related by passage from real to purely imaginary time. Renewed interest in pioneering work by Masuda, in particular, has further explored the option to circumnavigate blow-up in real time, by a detour in complex time.
In the present paper, the simplest scalar ODE case is studied for polynomials \begin{equation*} \label{*} %\displaywidth=155mm \dot{w}=f(w)=(w-e_0)\cdot\ldots\cdot(w-e_{d-1}), \tag{$*$} \end{equation*} of degree $d$ with $d$ simple complex zeros. The explicit solution by separation of variables and explicit integration is an almost trivial matter.
In a classical spirit, indeed, we describe the complex Riemann surface $\mathcal{R}$ of the global nontrivial solution $(w(t),t)$ in complex time, as an unbranched cover of the punctured Riemann sphere $w\in\widehat{\mathbb{C}}_d:=\widehat{\mathbb{C}}\setminus\{e_0,\ldots,e_{d-1}\}$. The flow property, however, fails at $w=\infty\in\widehat{\mathbb{C}}_d$. The global consequences depend on the period map of the residues $2\pi\mathrm{i}/f'(e_j)$ of $1/f$ at the punctures, in detail. We therefore show that polynomials $f$ exist for arbitrarily prescribed residues with zero sum. This result is not covered by standard interpolation theory.
Motivated by the PDE case, we also classify the planar real-time phase portraits of \eqref{*}. Here we prefer a Poincaré compactification of $w\in\mathbb{C}=\mathbb{R}^2$ by the closed unit disk. This regularizes $w=\infty$ by $2(d-1)$ equilibria, alternately stable and unstable within the invariant circle boundary at infinity. In structurally stable hyperbolic cases of nonvanishing real parts $\Re f'(e_j)\neq 0$, for the linearizations at all equilibria $e_j$, and in the absence of saddle-saddle heteroclinic orbits, we classify all compactified phase portraits, up to orientation-preserving orbit equivalence and time reversal. Combinatorially, their source/sink connection graphs correspond to the planar trees of $d$ vertices or, dually, the circle diagrams with $d-1$ nonintersecting chords. The correspondence provides an explicit count of the above equivalence classes of ODE \eqref{*}, in real time.
We conclude with a discussion of some higher-dimensional problems. Not least, we offer a 1,000 € reward for the discovery, or refutation, of complex entire homoclinic orbits.

In investigating dynamical systems with chaotic attractors, many aspects of global behavior of a flow or a diffeomorphism with such an attractor are studied by replacing a nontrivial attractor by a trivial one [1, 2, 11, 14]. Such a method allows one to reduce the original system to a regular system, for instance, of a Morse – Smale system, matched with it. In most cases, the possibility of such a substitution is justified by the existence of Morse – Smale diffeomorphisms with partially determined periodic data, the complete understanding of their dynamics and the topology of manifolds, on which they are defined. With this aim in mind, we consider Morse – Smale diffeomorphisms $f$ with determined periods of the sink points, given on a closed smooth 3-manifold. {We have shown that, if the total number of these sinks is $k$, then their nonwandering set consists of an even number of points which is at least $2k$. We have found necessary and sufficient conditions for the realizability of a set of sink periods in the minimal nonwandering set. We claim that such diffeomorphisms exist only on the 3-sphere and establish for them a sufficient condition for the existence of heteroclinic points. In addition, we prove that the Morse – Smale 3-diffeomorphism with an arbitrary set of sink periods can be implemented in the nonwandering set consisting of $2k+2$ points. We claim that any such a diffeomorphism is supported by a lens space or the skew product $\mathbb S^2\;\tilde{\times}\;\mathbb S^1$.

S. Smale has shown that any closed smooth manifold admits a gradient-like flow, which is a structurally stable flow with a finite nonwandering set. Polar flows form a subclass of gradient-like flows characterized by the simplest nonwandering set for the given manifold, consisting of exactly one source, one sink, and a finite number of saddle equilibria. We describe the topology of four-dimensional closed manifolds that admit polar flows without heteroclinic intersections, as well as all classes of topological equivalence of polar flows on each manifold. In particular, we demonstrate that there exists a countable set of nonequivalent flows with a given number $k \geqslant 2$ of saddle equilibria on each manifold, which contrasts with the situation in lower-dimensional analogues.

In this paper, we give a full description of all possible singular points that occur in generic 2-parameter families of vector fields on compact 2-manifolds. This is a part of a large project aimed at a complete study of global bifurcations in two-parameter families of vector fields on the two-sphere.

This study discusses an approach for estimation of the largest Lyapunov exponent for the mathematical model of the cardiovascular system. The accuracy was verified using the confidence intervals approach. The algorithm was used to investigate the effects of noises with different amplitudes and spectral compositions on the dynamics of the model. Three sets of parameters are considered, corresponding to different states of the human cardiovascular system model. It is shown that, in each case, the model exhibited chaotic dynamics. The model gave different responses to the changes in the characteristics of the noise, when using different sets of parameters. The noise had both constructive and destructive effects, depending on the parameters of the model and the noise, by, respectively, amplifying or inhibiting the chaotic dynamics of the model.

We study hyperchaotic attractors characterized by three positive Lyapunov exponents in numerical experiments. In order to possess this property, periodic orbits belonging to the attractor should have a three-dimensional unstable invariant manifold. Starting with a stable fixed point we describe several bifurcation scenarios that create such periodic orbits inside the attractor. These scenarios include cascades of alternating period-doubling and Neimark – Sacker bifurcations which, as we show, naturally appear near the cascade of codimension-2 period-doubling bifurcations, when periodic orbits along the cascade have multipliers $(-1, e^{i \phi}, e^{-i \phi})$. The proposed scenarios are illustrated by examples of the threedimensional Kaneko endomorphism and a four-dimensional Hénon map.