Backward Continuation Research Articles

Fast generation of collision-free low-thrust trajectories for asteroid landing using deep neural networks

This study presents a fast generation method of time-optimal low-thrust trajectories based on deep neural networks (DNN) for the collision-free asteroid landings. An equivalent unconstrained differentiable problem is transformed via the introduction of smooth penalty function, so that it can be addressed with the optimal control framework. To efficiently solve the equivalent problem to generate sufficient dataset for training DNNs, a double homotopy method is developed. Specifically, the equivalent problem is connected to the time-optimal control problem without anti-collision through two consecutive simplifications, and a solving process with double backward continuation is also formulated. Besides, two DNN models are constructed to provide high-quality predictions for the minimum glide-slope constraint angle of an initial state, and the initial costates and landing time for time-optimal low-thrust trajectories with anti-collision constraints, respectively. As such, the efficiency of collision-free low-thrust landing trajectory generation is improved significantly. For the cases within the range of training dataset or within a certain extension, the low-thrust landing trajectories with a very small terminal landing error can be generated using the predicted initial costates. Meanwhile, with the high-quality initial costates, the high-precision landing trajectories can be obtained by the double homotopy method in only a few iterative steps. Finally, the promising results in a Bennu collision-free asteroid landing scenario validate the efficiency of the proposed method.

Rayleigh–Bénard Convection With Multiple Solutions in Trapezoidal Closed Cavities

Abstract Rayleigh–Bénard convection (RBC) in symmetric trapezoidal closed cavities with cavity angle ϕ=70°−110°, filled with air, is studied using numerical simulations where inclined side walls are adiabatic. In contrast to rectangular cavities, where no flow exists below a threshold value, there is a weak convection even at a low Rayleigh number (Ra) due to the fact that there is a component of thermal gradient in the horizontal direction in these cavities. Interestingly, these cavities show sudden and significant jumps in the convection, similar to square cavities (Rac = 2585.02 for ϕ=90°), as Ra increases beyond a critical value (Rac). It is noted here that these Rac represent symmetry-breaking pitchfork bifurcations. These bifurcations are seen in both acute (Rac = 8000 for ϕ=70°) and obtuse (Rac = 2300 for ϕ=110°) angle trapezoidal cavities. Moreover, it is observed that multiple steady-state solutions (MSSS) exist as Ra is further increased. A forward and backward continuation approach for numerical simulations is used to track the co-existence of MSSS. These steady-states have co-existing one-roll and two-roll convective patterns beyond another threshold value of Ra. Here, two types of critical Ra have been identified for different cavity angles; one shows the sudden jump in the convection, and the other is the one beyond which MSSS co-exist. Furthermore, a codimension two bifurcation analysis is carried out with Ra and ϕ as two parameters. The bifurcation analysis divides the parameter space into different regions based on the multiplicity of the solutions.

Discrete hybrid Izhikevich neuron model: Nodal and network behaviours considering electromagnetic flux coupling

We analyse the dynamics of the improved discretised version of the well known Izhikevich neuron model under the action of external electromagnetic field. It is found that the improved three dimensional IZH map shows rich dynamics. With the variation of the electromagnetic field, period-doubling route to chaos in a repeating fashion is observed from the bifurcation diagram. Even the forward and backward continuation bifurcation diagram which do not completely overlap suggests that there is multistability in the system. The phenomenon of bistability (coexistence of periodic and chaotic attractors) is observed. The presence of periodic and chaotic attractor is aided by the maximal Lyapunov exponent diagram. The Lyapunov phase diagram of electromagnetic field and synapses current shows a large parameter region of chaotic and periodic behaviours with the presence of unbounded regions as well. The IZH map shows a plethora of spiking and bursting patterns such as mixed-mode patterns, tonic spiking, phasic spiking, steady spikes, regular spikes, spike bursting, periodic bursting, phasic bursting, chaotic firing etc with the variation of electromagnetic coupling strength and the synapses current. We also investigate the presence of chimera states in a ring-star, ring, star networks of IZH map neurons. Chimera states are found in the case of ring-star and ring network while synchronised clusters were found in the case of star network and are aided by the spatiotemporal plots, space-time plot, recurrence plots. The rich dynamics shown by the discretised IZH map makes it a promising research model to study about neurodynamics.

Antimonotonicity, Bifurcation and Multistability in the Vallis Model for El Niño

In this paper, the well-known Vallis model for El Niño is analyzed for the parameter condition [Formula: see text]. The conditions for the stability of the equilibrium points are derived. The condition for Hopf bifurcation occurring in the system for [Formula: see text] and [Formula: see text] are investigated. The multistability feature of the Vallis model when [Formula: see text] is explained with forward and backward continuation bifurcation plots and with the coexisting attractors. The creation of period doubling followed by their annihilation via inverse period-doubling bifurcation known as antimonotonicity occurrence in the Vallis model for [Formula: see text] is presented for the first time in the literature.

No Chattering and Adaptive Sliding Mode Control of a Fractional-Order Phase Converter with Disturbances and Parameter Uncertainties

Discussing the dynamical properties of various power system models is of significant importance in order to understand its complete behavior. Even though there are many literatures discussing about the chaotic behavior shown by phase converter circuits, none of them have reported the hazardous phenomenon of multistability. In this paper, we derive the fractional-order model of a phase converter circuit and investigate its dynamics. Bifurcation of the system with the parameters and fractional order are investigated. A forward and backward continuation scheme is adopted to display various coexisting attractors; the property of multistability is also discussed. Using forward and backward continuation, various coexisting attractors and the property of multistability are discussed. Two different sliding mode controllers for controlling chaotic oscillations with model disturbances and parameter uncertainties are derived, and the effectiveness of the controllers is discussed with numerical simulations.

Anisotropic Reverse-Time Migration of Ground-Penetrating Radar Data Collected on the Sand Dunes in the Badain Jaran Desert

Ground-penetrating radar (GPR) profiling is the primary tool to provide detail information of the internal structure and characteristics inside a sand dune in the desert area. However, with the severe elevation change in a short horizontal distance and on the rugged and complex surface topography of the sand dunes, getting clear imaging of the internal structure of sand dunes from the GPR profiling is still a big challenge. The classic imaging technique such as the Kirchhoff migration has been applied to process GPR data on sand dunes, with limited success. The reverse-time migration (RTM) technique is the most advanced imaging technique that can handles GPR data acquired on rugged surface with severe topographic relief to generate subsurface structural images with high fidelity and has been tried to process GPR data on sand dunes. The results are encouraging, and the imaging quality is significantly improved. The finite-difference time-domain (FDTD) method is the major numerical tool for forward and backward continuations of the wave field during the RTM process. There are two aspects for using FDTD in RTM of GPR data on sand dunes still need improvement: the numerical scattering caused by the staircase approximation of the ground surface by using gridding in the Cartesian coordinate, and the negligence of the possible anisotropy of the electromagnetic material properties due to the calcareous cementation bedding inside a sand dune. In this paper, we develop the RTM algorithm based on the staggered grid FDTD that handles the rugged topographic surface by using the curvilinear coordinate, and the possible anisotropic radar wave velocity of the sand dune media. We first demonstrate the equivalency of the nonuniform, isotropic medium and the uniform, anisotropic medium for justifying using the uniform, anisotropic velocity in the RTM by synthetic modeling. Next, we validate our approach of using the synthetic data with the comparison of using the Cartesian coordinate and the curvilinear coordinate in an isotropic medium. The results indicate that the RTM algorithm using the curvilinear coordinate can efficiently eliminate the adverse effect of the staircase approximated boundary of the topography surface. Finally, we processed the real GPR data collected on a sand dune in the Badain Jaran desert by using the curvilinear coordinate and the uniform, anisotropic velocity in FDTD forward and backward wave field continuation. Comparison of the results indicates that the RTM imaging using the boundary-conforming curvilinear coordinate and anisotropic velocity gains more coherent and higher resolution images for the calcareous cementation layers and the water table.

Transition to synchronization in a Kuramoto model with the first- and second-order interaction terms

We investigate a Kuramoto model incorporated with the first-order and the second-order interaction terms. We show that the model displays the coexistence of multiattractors and different attractors may be characterized by the phase distributions of oscillators. By investigating the transition diagrams in both forward continuation and backward continuation, we find that the synchronous state with unimodal phase distribution is the most stable one while the state in cluster synchrony with evenly distributed bimodal phase distribution is the least stable one. We also present the phase diagram of the model in the parameter space.

Hazardous Continuation Backward in Time in Nonlinear Parabolic Equations, and an Experiment in Deblurring Nonlinearly Blurred Imagery.

Identifying sources of ground water pollution, and deblurring nanoscale imagery as well as astronomical galaxy images, are two important applications involving numerical computation of parabolic equations backward in time. Surprisingly, very little is known about backward continuation in nonlinear parabolic equations. In this paper, an iterative procedure originating in spectroscopy in the 1930’s, is adapted into a useful tool for solving a wide class of 2D nonlinear backward parabolic equations. In addition, previously unsuspected difficulties are uncovered that may preclude useful backward continuation in parabolic equations deviating too strongly from the linear, autonomous, self adjoint, canonical model.This paper explores backward continuation in selected 2D nonlinear equations, by creating fictitious blurred images obtained by using several sharp images as initial data in these equations, and capturing the corresponding solutions at some positive time T. Successful backward continuation from t=T to t = 0, would recover the original sharp image. Visual recognition provides meaningful evaluation of the degree of success or failure in the reconstructed solutions.Instructive examples are developed, illustrating the unexpected influence of certain types of nonlinearities. Visually and statistically indistinguishable blurred images are presented, with vastly different deblurring results. These examples indicate that how an image is nonlinearly blurred is critical, in addition to the amount of blur. The equations studied represent nonlinear generalizations of Brownian motion, and the blurred images may be interpreted as visually expressing the results of novel stochastic processes.

The Influence of Time Shift Wavelet on the Migration Imaging Accuracy

The imaging problems of common shot data with prestack reverse time migration method are solved with the forward continuation of source wave field and the backward continuation of receiver wave field in depth direction at the same time. In order to obtain imaging value, the conventional method is that the two wave fields implement cross correlation at the same time and stack the cross correlation value at all the same time. This method is fit for the case where the seismic wavelet is a pulse or minimum phase. However, in generally speaking, the event picked from seismic record is the peak of seismic wavelet which has a short time shift relative to the wavelet initial time. It subsequently results in some imaging position errors in cross correlation method. In this paper, theoretical seismic record are simulated and migrated by wave equation finite element prestack reverse time migration method with cross correlation imaging conditions. The influence of the time shift wavelet both in seismic source wave field and receiver wave field on the imaging accuracy is discussed. Numerical experiments demonstrate that the influence of time shift wavelet on the position error is related to the relative delay length of the peak of wavelet in the two wave fields and the medium velocity around the reflection point. The results provide a theoretical reference for the selection of seismic wavelet, which is important to the application of the cross correlation imaging conditions in prestack reverse time migration method.

On the linear instability of a finite Stokes layer: Instantaneous versus Floquet modes

The linear instability of a finite Stokes layer is investigated numerically. The aim is to clarify the relationship between the instantaneous instability and the recently discovered Floquet global instability [P. J. Blennerhassett and A. P. Bassom, J. Fluid Mech. 464, 394 (2002); P. J. Blennerhassett and A. P. Bassom, J. Fluid Mech. 556, 1 (2006)], and their role in causing transition to turbulence. For the first time, a numerical approach based on initial-value calculations is able to confirm the global instability, which occurs as the Reynolds number R (based on the Stokes-layer thickness) exceeds the critical value Rc=708. In both supercritical (R>Rc) and subcritical (R<Rc) regimes, strong instantaneous growth occurs during the deceleration phase. By projecting the disturbance at an arbitrary instant to instantaneous eigenfunctions, it is shown that the energy of the disturbance is primarily carried by the most unstable instantaneous mode. During the main growing phase, both the spatial structure and the growth rate agree well with those of the instantaneous mode; during the decaying phase part of the energy is continuously transferred to the precursor (i.e., the backward continuation in time) of the most unstable mode existing in the next cycle. The most unstable mode including its continuation therefore constitutes the crucial part of a Floquet mode. Unstable Floquet modes exist despite that none of instantaneous modes is periodic, implying that the commonly held premise that a Floquet mode is equivalent to a periodic instantaneous mode is erroneous. It is also found that in the subcritical regime, the flow responds very sensitively to small surface roughness, modeled by a wavy wall. The sensitivity can be measured by a response function, and our calculations show that significant disturbances can be generated by surface imperfections less than 0.1 μm in typical experimental setups. The response becomes stronger as R increases, and tends to infinity as the wavenumber of the wavy wall approaches that of the critical Floquet mode at Rc, indicating that roughness with a suitable wavelength can excite a global mode through resonance, and the large response may be attributed to a detuned resonance with a global mode.

Continuous dependence on initial geometry in linear elastodynamics on a half-cylinder

Continuous dependence on the initial time geometry is established for the mean-square integral of the displacement in a linear inhomogeneous anisotropic elastic semi-infinite cylinder in motion subject to a prescribed time-dependent base displacement and initial data. A bound, newly derived for the total energy, in conjunction with backward continuation in time of the unperturbed and perturbed displacements, is employed to obtain the result.

Sturmian nodal set analysis for higher-order parabolic equations and applications

We describe the local pointwise structure of multiple zeros of solutions of $2m$th-order linear uniformly parabolic equations, \begin{equation} \tag*{(0.1)} u_t = \sum _{|{\beta} | \le 2m} \, a_{\beta}(x,t) D^{\beta}_x u \quad {\rm in} \,\,\, {{\bf R}^N} \times [-1,1] \quad (m \ge 2), \end{equation} with bounded and Lipschitz-continuous (for $|{\beta}|=2m$) coefficients, in the existence-uniqueness class $\{|u(x,t)| \le B {\mathrm e}^{b|x|^{\alpha}}\}$, where $B,b>0$ are constants and ${\alpha} = \frac {2m}{2m-1}$. Assuming that $u(0,0)=0$ and using the Sturmian backward continuation blow-up variable $ y=x/(-t)^{\frac 1{2m}} \quad (t < 0), $ we perform a classification of all possible types of formation as $t \to 0^-$ of multiple spatial zeros of the solutions $u(x,t)$. We show that there exists a countable family of multiple zeros evolving as $t \to 0^-$ according to the nodal sets of polynomial eigenfunctions of a non-self-adjoint operator ${{\mathbf B}}^*$ associated with that in (0.1). Next, we show that other related polynomial solutions occur in the collapse of multiple zeros as $t \to 0^+$, which is described in terms of the forward continuation variable $ y=x/t^{\frac 1{2m}} \quad (t>0). $ For the 1D second-order ($m=1$) parabolic equation with smooth coefficients, $$ u_t= a(x,t) u_{xx} + q(x,t) u \quad (a(x,t) \ge a_0>0), $$ this two-step analysis is known as {Sturm's Second Theorem} on zero sets, established by C.~Sturm in 1836, [32]. His more famous First Theorem (the number of zeros of solutions is non-increasing with time ) was derived as a consequence of the second one. In the last thirty years these PDE ideas of Sturm found new applications, generalizations and extensions in various areas of general parabolic theory, stability and orbital connection problems, unique continuation and Poincaré--Bendixson theorems, mean curvature and curve shortening flows, symplectic geometry, etc. Using such a local classification of multiple zeros, we establish a unique continuation theorem for higher-order parabolic PDEs and inequalities, and estimate the Hausdorff dimension of nodal sets of solutions. It turns out that some common features of multiple zeros formation can be observed for solutions of other PDEs including linear dispersion and wave equations, $$ u_t=u_{xxx} \quad \mbox{and} \quad u_{tt}=u_{xx}, $$ and we present a discussion on this issue.

Collapsible Backward Continuation and Numerical Approximations in a Functional Differential Equation

The phenomenon of a wild kind of oscillatory behavior found in solutions of the scalar differential equation with scaling on time,x(t)=−ax(t)+ax(pt),is investigated in this paper. It is shown that this phenomenon does not, in general, reflect a similar behavior in this equation's numerical solutions. The results obtained here may be extended, under appropriate assumptions, to other types of nonlinear multidimensional retarded functional differential equations with vanishing delays.

A palaeomagnetic study of Svecokarelian basic rocks from northern Sweden

Abstract This palaeomagnetic investigation comprises basic rocks from six localities from the Svecokarelian zone in northern Sweden. Most of the pole positions in this study and other reported poles of Svecokarelian and post-Svecokarelian rocks fall within an approximately 12 degrees wide band running from east to west representing ages of magnetizations from 1880-1700 to ∼ 1530 Ma. Thermal demagnetizations of specimens of the probably oldest massifs indicate a possible backward continuation of the polar wandering path. Mineralogical studies of thin sections of the rocks show ore symplectites and myrmekitic textures indicating a slow rate of cooling at least at the end of the rock formation. Signs of metamorphism are demonstrated by the existence of secondary minerals, including magnetite, not related to late magmatic alterations. The distribution of site means as well as the change of directions of the remanence vectors during thermal demagnetization can be explained by a slow rate of cooling and where signs of metamorphism exist by partial remagnetization of the rock. This study has, apart from the palaeomagnetic results, demonstrated the difficulty of correlating radiometric ages with ages of magnetization.

The nervous system of Loligo. II. Suboesophageal centres.

A well-marked hierarchy of centres can be recognized within the suboesophageal lobes and ganglia of the arms. The inputs and outputs of each lobe are described. There are sets of motoneurons and intermediate motor centres, which can be activated either from the periphery or from above. They mostly do not send fibres up to the optic or higher motor centres. However, there is a large set of fibres running from the magnocellular lobe to all the basal supraoesophageal lobes. The centre for control of the four eye-muscle nerves in the anterior lateral pedal lobe receives many fibres direct from the statocyst and from the peduncle and basal lobes, but none direct from the optic lobe. The posterior lateral pedal is a backward continuation of the oculomotor centre, containing large cells that may be concerned in initiating attacks by the tentacles. An intermediate motor centre in the posterior pedal lobe probably controls steering. It sends fibres to the funned and head retractors, and by both direct and interrupted pathways to the fin lobe. It receives fibres from the crista nerve and basal lobes, but none direct from the optic lobe. The jet control centre of the ventral magnocellular lobe receives fibres from the statocyst and skin and also from the optic and basal lobes. Some of these last also give extensive branches throughout the palliovisceral lobes. The branching patterns of the dendritic collaterals differ in the various lobes. Some estimates are given of the numbers of synaptic points. The dendritic collaterals of the motoneurons spread through large volumes of neuropil and they overlap. The incoming fibres spread widely and each presumably activates many motoneurons either together or serially. Many of the lobes contain numerous microneurons with short trunks restricted to the lobe, but there are none of these cells in the chromatophore lobes or fin lobes. The microneurons have only few dendritic collaterals, in contrast to the numerous ones on the nearby motoneurons.

Backward continuation of retarded functional differential equations

Backward continuation of retarded functional differential equations

Forward and backward continuation for neutral functional differential equations

Basic theory of existence, uniqueness, and continuation for neutral functional differential equations

THE MOBILE RIGHT COLON

In 20 per cent of persons there occurs, as the result of a fault in embryologic development, an absence of fixation of the ascending colon to the posterior abdominal wall. This absence of fixation may concern the entire right colon or may be limited to the pendulous cecum. The mobility of the right colon is often very great. When examined under the screen, the vertical excursion of the hepatic flexure as between the upright and the recumbent postures may be as much as 6 or 7 inches. It is often possible to push the whole of the right colon over to the left of the spine. Instead of being fixed, the right colon is found to possess a mesentery of varying length and, in addition to the true mesentery, there usually develops a backward continuation and elongation of it, the pseudomesentery, a firm dense structure coming from the fascia of