The nonlinear Schrödinger equation possesses doubly periodic solutions expressible in terms of the Jacobi elliptic functions. Such solutions can be realized through doubly periodic patterns observed in experiments in fluid mechanics and optics. Stability and robustness of these doubly periodic wave profiles in the focusing regime are studied computationally by using two approaches. First, linear stability is considered by Floquet theory. Growth will occur if the eigenvalues of the monodromy matrix are of a modulus larger than unity. This is verified by numerical simulations with input patterns of different periods. Initial patterns associated with larger eigenvalues will disintegrate faster due to instability. Second, formation of these doubly periodic patterns from a tranquil background is scrutinized. Doubly periodic profiles are generated by perturbing a continuous wave with one Fourier mode, with or without the additional presence of random noise. Effects of varying phase difference, perturbation amplitude, and randomness are studied. Varying the phase angle has a dramatic influence. Periodic patterns will only emerge if the perturbation amplitude is not too weak. The growth of higher-order harmonics, as well as the formation of breathers and repeating patterns, serve as a manifestation of the classical problem of Fermi-Pasta-Ulam-Tsingou recurrence.

The increasing number and variety of spacecraft that are expected to operate within cislunar space and other multi-body gravitational environments throughout the solar system necessitates the continued development of strategies for rapid trajectory design and design space exploration. In the field of robotics, similar needs have been addressed using motion primitives that capture the fundamental building blocks of motion and are used to rapidly construct complex paths. Inspired by this concept, this paper leverages motion primitives to construct a framework for rapid and informed spacecraft trajectory design in a multi-body gravitational system. First, motion primitives of fundamental solutions, e.g., selected periodic orbits and their stable and unstable manifolds, are generated via clustering to form a discrete summary of segments of the phase space. Graphs of motion primitives are then constructed and searched to produce primitive sequences that form candidate initial guesses for transfers of distinct geometries. Continuous transfers are computed from each initial guess using multi-objective constrained optimization and collocation. This approach is demonstrated by constructing an array of geometrically distinct transfers between libration point orbits in the Earth-Moon circular restricted three-body problem with impulsive maneuvers.

Floquet multipliers determine the local asymptotic stability of a periodic solution and, in the context of parameter dependence, determine also its bifurcations. This paper deals with numerical aspects of the computation of the Floquet multipliers for three classes of functional differential equations: ordinary differential equations (ODEs), differential equations with constant delay (DDEs) and differential equations with state-dependent delay (sd-DDEs). Using a collocation approach for computing periodic solutions, we obtain an approximation of the (corresponding) monodromy operator, a monodromy matrix. The eigenvalues of this matrix form an approximation to the Floquet multipliers. The accuracy of the computed multipliers is an important issue in bifurcation analysis of a dynamical system. As far as we know, no prior work on the study of the convergence and accuracy of computed Floquet multipliers for DDEs and sd-DDEs exists. We analyze the dependency of the accuracy of the computed multipliers on the parameters and on the type of collocation approximation. In particular, we show that the accuracy of the computed trivial multiplier is not always comparable to the accuracy of the computed periodic solution and the accuracy of the other computed multipliers.

This work presents the application of Lyapunov Characteristic Exponents (LCEs), or in short Lyapunov Exponents, to the evaluation of rotorcraft aeroelastic stability. Current state of art literature on rotorcraft aeroelastic stability analysis approaches the problem by either using a constant coefficient approximation or by computing the eigenvalues of the monodromy matrix according to Floquet Theory. The former neglects periodicity and the latter is only applicable to the perturbation of the problem about a periodic orbit. Often such approximations are acceptable; however, LCEs can be applied to generic trajectories of non-linear systems to produce an estimate of the stability properties without the need to reach a steady orbit or determine the period of the system. Being more general, LCEs can provide a common environment in rotorcraft aeroelastic stability among both linear and non-linear systems, and be applicable to all problems that can be proficiently analyzed by time marching analysis, including experimental data. This work presents the evaluation of the stability of an isolated rotor formulated as a linear time-periodic system by computing the LCEs from arbitrary fiducial trajectories. The method is illustrated in relation with the problem of rigid helicopter blade flapping, and ground resonance with one damper inoperative and with non-linear dampers.

SUMMARY We present an optimal gain scheduling control design for bipedal walking with minimum tracking error. We obtained a linear approximation by linearizing the nonlinear hybrid dynamic model about a nominal periodic trajectory. This linearization allows us to identify the linear model as a linear periodic system. An optimal feedback was designed using Bellman's dynamic programming. The linear periodic system allows us to determine a linear quadratic regulator (LQR) for a single period and to set the Hamilton-Jacobi-Bellman (HJB) function in a linear quadratic form. In this way, the dynamic programming yielded an admissible continuous gain scheduling that was designed with regard to the hybrid dynamics of the system. We tuned the optimization parameters such that the tracking error and the average energy consumption are minimized. Due to linearization, we were able to examine the stability of the approximated periodic system achieved by the periodic gain according to Floquet's theory, by calculating the monodromy matrix of the closed-loop hybrid system. In addition to determining stability, the eigenvalues of this approximated monodromy matrix allowed us to evaluate the settling time of the system. This approach presents a direct method for optimal solution of locomotion control according to a given reference trajectory.

Prediction of dynamical stability of piecewise linear oscillators’ responses significantly depends on very small harmonic terms of the actual time domain response. The influence of these small harmonic terms on convergence of piecewise linear oscillators’ monodromy matrix eigenvalues (that determine the dynamical stability of the steady-state response) is considered in this paper. For this purpose, a simple one degree-of-freedom system with a piecewise-linear force-displacement relationship subjected to a harmonic excitation is analysed. Stability of the periodic response obtained in the frequency domain by the incremental harmonic balance method is determined by using the Floquet-Liapounov theorem. Responses in the time domain are obtained by a method of piecing the exact solutions. Using the frequency plot of maximum modulus of the eigenvalues of the monodromy matrix is proposed in order to make the stability and the bifurcation analysis of piecewise linear oscillators more reliable and efficacious.

Parallel controlled DC-DC converters are nonlinear and non-smooth systems, they show various nonlinear behaviour including smooth, non-smooth bifurcation, and chaos when they work outer their design conditions. Usually, the Poincaré map approach is the most common method for studying the stability of those nonlinear systems. Stability is indicated using the eigenvalues of the Jacobian of the map computed at the fixed point. The other method is the monodromy matrix approach, where the stability can be concluded by computed the eigenvalues of the matrix. In this paper, the nonlinear dynamics of parallel connected DC-DC converters are investigated. It is shown that the concept of the monodromy matrix can be applied to determine the stability of the system as well as the Poincare map approach.

Modular multilevel converters (MMCs) have been widely used in high-voltage direct current (HVDC) transmission systems, renewable energy and storage integration, static var compensation, and so on. The nonlinear time-periodic nature of MMCs leads to complex harmonic interactions and poses significant challenges to the small-signal stability analysis of grid-connected MMC systems. The Floquet theory-based monodromy matrix method can accurately achieve this task but requires time-consuming numerical integration to construct the matrix. To address this, an analytical monodromy matrix method is proposed in this paper for efficient and accurate small-signal stability analysis of grid-connected MMC systems. First, an analytical monodromy matrix of linear time-periodic (LTP) systems, i.e., the system's state transition matrix over a period, is efficiently constructed by applying the Chebyshev collocation method instead of numerical integration. In addition to low computational burden, the analytical feature of the matrix allows one to derive the derivative-based eigenvalue sensitivity. Then, the system free-response solution is efficiently computed by means of a set of analytical state transition matrices. By using the fast Fourier transform, the effective oscillation components in critical LTP modes and the related participation factors are identified. Both participation factor analysis and damping ratio analysis can be performed to gain insightful characterization of system dynamics. The correctness and effectiveness of the proposed method are verified on an exemplary grid-connected MMC system by both numerical and experimental results.

The Floquet exponents of periodic field lines are studied through the variations of the magnetic action on the magnetic axis, which is assumed to be elliptical. The near-axis formalism developed by Mercier, Solov'ev and Shafranov is combined with a Lagrangian approach. The on-axis Floquet exponent is shown to coincide with the on-axis rotational transform. A discrete solution suitable for numerical implementation is introduced, which gives the Floquet exponents as solutions to an eigenvalue problem. This discrete formalism expresses the exponents as the eigenvalues of a $6$ X $6$ matrix.

Fluid-fluid interfacial instability and subsequent fluid mixing are ubiquitous in nature and engineering. The hydrodynamic instability of fluid interfaces has long centered on the pressure gradient-driven long-wavelength Rayleigh-Taylor instability and the resonance-induced short-wavelength Faraday instability. However, neither instability alone can explain the dynamics when both mechanisms are present. We identify a previously unseen multi-modal instability emerging from their coexistence. When the denser fluid is polydimethylsiloxane, the mixed region at a high density contrast (Atwood number = 0.9) spans a vibration amplitude range approximately twice the gravitational acceleration. Using Floquet stability analysis, we show how vibrations govern transitions between the RT and Faraday instabilities, leading to contention between these instabilities rather than resonant enhancement. Here, the initial transient growth is represented by the exponential modal growth of the most unstable Floquet exponent, along with its accompanying periodic behavior. Direct numerical simulations validate these findings and track interface breakup into the multiscale and nonlinear regimes. Specifically, we show that growing RT modes nonlinearly suppresses Faraday responses even when the initial growth rate of the Faraday instability is 3.63 times that of RT, so a bidirectional competition hinders their sustained coexistence.

This paper explores the dynamic analogy between the discrete Lorenzian attractor and a modified Kropotov-Pakhomov neural network (MRNN). A one-dimensional peak map is used to extract the successive maxima of the Lorenzian system and preserve the basic properties of the chaotic flow. The MRNN, governed by the Bogdanov-Hebb learning rule with dissipative feedback, is formulated as a discrete nonlinear operator whose parameters can reproduce the same hierarchy of modes as the peak map. It is theoretically shown that the map multiplier and the spectral radius of the monodromy matrix of the MRNN provide equivalent stability conditions. Numerical diagrams confirm the correspondence between the control parameters of the Lorenz model and the network parameters. The results establish the MRNN as a neural emulator of the Lorenz attractor and offer an analysis of self-organization and stability in adaptive neural systems.

This paper uses a motion primitive approach to automatically generate constrained spacecraft trajectories for Neptunian system exploration. Motion primitives are generated as smaller building blocks of motion that summarize periodic orbits and arcs along stable and unstable manifolds of selected orbits in the Neptune-Triton circular restricted three-body problem. The sequential composability of these motion primitives is represented by a graph that also incorporates path and maneuver constraints. This graph is searched using a k-best paths algorithm to generate multiple motion primitive sequences. These sequences are transformed into an array of geometrically diverse initial guesses. After corrections and optimization, the resulting tradespace of continuous, constrained trajectories with impulsive maneuvers is analyzed. This approach is applied to two planar trajectory design scenarios in the Neptunian system: high-energy insertion into a Neptune-centered science orbit after interplanetary arrival and low-energy transfers between science orbits centered around each of Neptune and Triton.<h4>Supplementary information</h4>The online version contains supplementary material available at 10.1007/s40295-025-00545-z.