We study the chaotic dynamics of the Suslov top, a classical nonholonomic system. This system possesses a rich bifurcation structure, which we explore numerically via Lyapunov diagrams and bifurcation analysis in a two-parameter plane. This structure is organized by a codimension-2 point where an equilibrium state has eigenvalues $(0, i\omega,−i\omega)$ and a homoclinic loop. From this point, two important bifurcation curves originate: the Andronov – Hopf bifurcation curve, corresponding to the birth of a stable periodic orbit from an equilibrium state, and the Andronov – Vitt – Shilnikov bifurcation curve, corresponding to the birth of a stable periodic orbit from a homoclinic loop of a saddle-node equilibrium. We show how chaotic attractors and repellers emerge from equilibrium states, grow, and merge into mixed dynamics. We also discuss the connection of this organizing codimension-2 point with the blue sky catastrophe phenomenon: the birth of a stable periodic orbit of infinite period and length.

In this paper, we constructed a new model of the neuron membrane potential in the form of a piecewise-smooth (PWS) slow-fast system. Its fast part is a two-dimensional PWS subsystem specially “sewn” from the normal forms of a stable focus and a saddle-node bifurcation, and the slow one-dimensional equation serves to periodically force the fast one. Our approach to constructing PWS systems in this way allows them to qualitatively reproduce the phase space structure of their smooth counterpart and also provide explicit solutions. By obtaining an exact Poincaré map as a composition of solutions and examining it, we proved the complete bifurcation structure of the fast subsystem. By selecting three bifurcation routes and applying a time-periodic change of the subsystem parameter along them using the third slow equation, we obtained three different types of bursting patterns. Finally, we showed that, when coupled into a network, our PWS models form a synthetic central pattern generator that manifests a traveling bursting wave.

We discuss algebraic and combinatorial aspects of the Hamiltonian normal form theory. The main objective is to describe the normal form near a singular point by an explicit formula in terms of the original Hamiltonian, avoiding the normalization procedure. In the case of one degree of freedom we compute the normal form as an explicit nonlinear functional, applied to the original Hamiltonian. In arbitrary dimension the corresponding formulas are more complicated but still explicit.

We study Axiom A diffeomorphisms (in the sense of Smale) on a closed orientable $n$-manifold $M^n$, $n\geqslant 4$, whose nonwandering sets consist of codimension one orientable attractors, repellers, and isolated saddle periodic orbits with the same Morse index. Such systems combine chaotic dynamics on basic sets with regular dynamics near trivial periodic orbits. We prove that the supporting manifold $M^n$ is homeomorphic to a connected sum of several copies of the $n$-dimensional torus. Moreover, we obtain a complete classification of such systems up to global conjugacy on their nonwandering sets. The invariant of conjugacy is a finite collection of Anosov automorphisms of the torus, called a tube, defined up to equivalence under linear automorphisms with determinants of the same sign. This extends the classical Plykin – Grines classification of individual codimension one basic sets to systems with multiple coexisting attractors and repellers.

We consider the plane three-body problem with two of the masses much smaller than the third. Periodic solutions with near collisions of the small bodies were named by Poincaré second species solutions. Poincaré gave an idea of how to prove their existence, but didn’t provide an actual proof. Complete proofs appeared much later and only for the restricted three-body problem. We reduce the problem of second species solutions of the general threebody problem to the study of dynamics of a billiard type problem and show how to transform the arguments of Poincaré into a rigorous proof.

This paper extends the modeling and control of a spherical robot with electromagnetic drive (SRED) from planar to fully three-dimensional motion on uneven surfaces. The robot consists of two concentric spherical shells: an outer shell carrying permanent magnets (PMs) and an inner shell with solenoid-type electromagnets (EMs); rolling is produced by the controlled electromagnetic interaction between the two magnet arrays. To enable omnidirectional rolling, the magnet placement is generalized to vertex-transitive polyhedra with cubic or icosahedral symmetry, which preserves the scalarity of the inertia tensors and allows the existing optimal control framework to be applied without modification. Within this class, $M=60$ PMs on a rhombicosidodecahedron and $S=24$ EMs on a truncated octahedron are selected; numerical evaluation over $10^{4}$ relative orientations confirms that a nonzero torque can be generated about an arbitrary axis. The drive model is reformulated as a three-dimensional torque map matrix, and coil currents and terminal voltages are recovered from a reference torque via the generalized-$H_{2}$ optimal controller, a Tikhonov-regularized pseudoinverse, and Kirchhoff's circuit equations with back-EMF terms. Simulations of straight-line rolling, lateral-offset correction, and three-dimensional path tracking on a Gaussian bell-shaped surface confirm convergence of the tracking error to zero, with currents and voltages within the range of standard low-voltage motor drivers.

We study bifurcation scenarios leading to the so-called Shilnikov singular attractors as a result of breakdown of closed invariant curves in the Chialvo map, which is a twodimensional endomorphism demonstrating neuron-like dynamics. We show that two different routes, soft and hard, of the emergence of such closed invariant curves can be traced here: the soft one corresponds to the birth of an invariant curve as a result of a supercritical Neimark – Sacker bifurcation, and the hard one relates to the immediately occurring big invariant curve after disappearance of a stable fixed point at the saddle-node bifurcation. We study both these mechanisms and trace subsequent scenarios of breakdown of the invariant curve and chaos development leading to the emergence of Shilnikov singular attractors. Additionally, we study geometrical peculiarities of these attractors, such as structures of rotating patterns of orbits inside the Shilnikov singular funnel and present a two-parametric analysis with Lyapunov exponents and minimal a distance between the chaotic attractor and the unstable focus.

In the circular restricted three-body problem, families of periodic and quasi-periodic orbits exist in the vicinity of the libration points and are of significant interest for practical applications. This work presents a neural network-based algorithm for orbit construction and station-keeping, applicable to both periodic and quasi-periodic orbits near collinear libration points. The method employs a fully connected neural network trained on labeled trajectory data and iteratively refined using trajectories generated during previous stages of the algorithm. The proposed approach was applied to orbit generation near the Sun-Earth L1 libration point and demonstrated the ability to generalize to a broader class of trajectories than those included in the training set.

Strange nonchaotic attractors (SNA) are fractal invariant sets characterized by aperiodic dynamics and nonpositive largest Lyapunov exponents. Due to the fractal geometry and the absence of exponential sensitivity to initial conditions, SNA are often difficult to distinguish from chaotic attractors using time-series data. This study examines the applicability of wavelet-based analysis for the detection of strange nonchaotic regimes in dynamical systems. The proposed approach involves the analysis of wavelet scalograms and global wavelet energy spectra. It is shown that quasiperiodic and strange regimes can be distinguished by the structure of their scalograms. To distinguish between strange nonchaotic and chaotic regimes, global wavelet energy spectra are employed. Numerical results demonstrate the applicability of this method for identifying quasiperiodic, strange nonchaotic, and chaotic regimes from time-series data.

This paper treats the problem of finding feasible periodic motions and that of their robust orbital stabilization for a spherical robot. The first problem is tackled using the servoconstraint approach based on the mathematical model of a pendulum-type spherical robot moving on a plane under the no-slip and no-loss-of-contact assumptions. The second problem is addressed through a generalization of the well-known Andronov – Vitt theorem, specifically by introducing the transverse coordinates of co-dimension one that are zero on a desired orbit. Linearizing the dynamics in the transverse coordinates results in a linear system with periodic coefficients (LSPC). A rigorous procedure is suggested for stabilizing the LSPC in the presence of structured uncertainties and matched disturbances using vector relay control. This control strategy forces the LSPC solutions onto a predefined time-periodic family of subspaces, where the exponential stability is ensured. The effectiveness of the overall framework is demonstrated through simulations.