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,iω,-iω) 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 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.
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.
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.
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.
