Stable Periodic Orbits Research Articles

Transfers to Earth-Moon triangular libration points by Sun-perturbed dynamics

This paper is devoted to the construction of transfers to Earth-Moon triangular libration points based on their Sun-perturbed dynamics in the Sun-Earth-Moon system. In this bi-circular problem (BCP), the solar gravity perturbation breaks the stability of the Earth-Moon triangular libration points with three remaining periodic orbits (two stable and one unstable) around them as their dynamical substitutions. As the stable region, the invariant tori full of quasi-periodic orbits around the two stable periodic orbits can be used as the destination of triangular libration point transfers for a low station-keeping cost. However, the instability of the unstable periodic orbit is so mild that its stable and unstable manifolds are folded around the invariant tori surrounding the other two stable orbits. Thus, the authors approximate the stable and unstable manifolds of the unstable orbit through the parameterization method, which emerge from the Sun-perturbed Earth-Moon triangular libration points vicinity. They act as a bridge between outer space and the stable region around triangular libration points. First, the transfers connecting the Earth’s vicinity are constructed using the manifolds. Subsequently, a new connection mechanism near L4 and L5 based on the N:M multi-loop transfers is developed by extending the conventional patching skill to time-dependent situations. The quickest one is achieved by a 1:1 L4-L5 heteroclinic-based connection with a TOF of less than 26.9 days at a fuel cost of 117.88 m/s. Note that such Sun-perturbed transfers cannot be produced by continuing any trajectory in the Earth-Moon circular restricted three-body problem but they can generate two-impulsive transfers in the ephemeris model by continuation.

Combinatorics of linear stability for Hamiltonian systems in arbitrary dimension

We address the general problem of studying linear stability and bifurcations of periodic orbits for Hamiltonian systems of arbitrary degrees of freedom. We study the topology of the GIT sequence introduced by the first author and Urs Frauenfelder in [7], in arbitrary dimension. In particular, we note that the combinatorics encoding the linear stability of periodic orbits is governed by a quotient of the associahedron. Our approach gives a topological/combinatorial proof of the classical Krein–Moser theorem, and refines it for the case of symmetric orbits.

Ballistic to diffusive transition for swimmers in a periodic vortex array.

We study the transport of rigid ellipsoidal swimmers in a periodic vortex array via numerical simulation and dynamical systems analysis. Via ensemble simulations, we show the counterintuitive result that slower swimming speeds can generate fast ballistic transport, while faster swimming speeds generate chaotic and diffusive transport, which is inherently slower in the long run. To explain this, we use the symmetry of the flow to construct a time-reversible Poincaré return map on a two-dimensional surface of sectionin phase space. For sufficiently small swimming speeds, we find stable periodic orbits on the surface of sectionsurrounded by invariant tori, similar to Kolmogorov-Arnold-Moser curves. Trajectories within these tori are ballistic. As the swimming speed is increased, the periodic orbits undergo a sequence of period-doubling bifurcations that destroys the ballistic tori. These bifurcations exactly match the ballistic to diffusive transition from the ensemble simulations. Additional ensemble simulations are used to test the robustness of these results to noise. The ballistic behavior is destroyed as the strength of rotational diffusion increases. However, we estimate that the ballistic tori might still be seen in experiments.

The role of dynamic friction in the appearance of periodic oscillations in mechanical systems

This article investigates the appearance of periodic mechanical oscillations associated with the transition between static and dynamic friction regimes. The study employs a mechanical system with one degree of freedom and a friction model recently proposed by Brown and McPhee, whose continuity and differentiability properties make it particularly appropriate for an analytical treatment of the equations. A bifurcation study of the system, including stability analysis, transformation to normal form and numerical continuation techniques, reveals that stable periodic orbits can be created either by a supercritical Hopf bifurcation or by a saddle-node bifurcation of limit cycles. The influence of all system parameters on the appearance of periodic oscillations is investigated in detail. In particular, the effect of the friction model parameters (static-to-dynamic friction ratio and transition speed between the static and dynamic regimes) on the bifurcation behavior of the system is addressed.

Mathematical Structure of RelB Dynamics in the NF-κB Non-Canonical Pathway

This study analyzed the non-canonical NF-κB pathway, which controls functions distinct from those of the canonical pathway. Although oscillations of NF-κB have been observed in the non-canonical pathway, a detailed mechanism explaining the observed behavior remains elusive, owing to the different behaviors observed across cell types. This study demonstrated that oscillations cannot be produced by the experimentally observed pathway alone, thereby suggesting the existence of an unknown reaction pathway. Assuming this pathway, it became evident that the oscillatory structure of the non-canonical pathway was caused by stable periodic orbits. In addition, we demonstrated that altering the expression levels of specific proteins reproduced various behaviors. By fitting 14 parameters, excluding those measured in previous studies, this study successfully reproduce nuclear retention (saturation), oscillation, and singular events that had been experimentally confirmed. The analysis also provided a comprehensive understanding of the dynamics of the RelB protein and suggested a potential inhibitory role for the unknown factor. These findings indicate that the unknown factor may be an isoform of IκB, contributing to the regulation of NF-κB signaling. Based on these models, we gained invaluable understanding of biological systems, paving the way for the development of new strategies to manipulate specific biological processes.

Comparison of Homoclinic Bifurcations Between Grid-Following and Grid-Forming Converters

A second-order model is developed to describe the essential behavior of the grid-following converter (GFLC) and the grid-forming converter (GFMC) for studying the synchronization characteristics of grid-connected converter systems. In this paper, a general set of criteria in terms of the manifolds and saddle quantity is derived for studying the homoclinic bifurcation behavior of GFLCs and GFMCs. It is shown that three unstable periodic orbits and two homoclinic bifurcations may exist for the GFLC, while only one stable periodic orbit and one homoclinic bifurcation may exist for the GFMC. It is also found that after the onset of the first homoclinic bifurcation, the stable equilibrium point (SEP) is surrounded by the stable manifolds of the unstable equilibrium point (UEP), which forms the SEP's basin of attraction, for both GFLCs and GFMCs. In this case, the converters may lose their synchronization if the trajectory crosses over the UEP. Therefore, system parameters should be designed to avoid the onset of the aforementioned bifurcation so that the SEP's basin of attraction can be significantly enlarged to contain the UEP. As a result, the GFLC or GFMC can eventually resynchronize with the grid even when its trajectory passes the UEP. Finally, experimental results are provided to verify these theoretical findings.

The Evolution of the Phase Space Structure Along Pitchfork and Period-Doubling Bifurcations in a 3D-Galactic Bar Potential

We investigate how the phase space structure of a Three-Dimensional (3D) autonomous Hamiltonian system evolves across a series of successive Two-Dimensional (2D) and 3D pitchfork and period-doubling bifurcations, as the transition of the parent families of Periodic Orbits (POs) from stability to simple instability leads to the creation of new stable POs. Our research illustrates the consecutive alterations in the phase space structure near POs as the stability of the main family of POs changes. This process gives rise to new families of POs within the system, either maintaining the same or exhibiting higher multiplicity compared to their parent families. Tracking such a phase space transformation is challenging in a 3D system. By utilizing the color and rotation technique to visualize the Four-Dimensional (4D) Poincaré surfaces of section of the system, i.e. projecting them onto a 3D subspace and employing color to represent the fourth dimension, we can identify distinct structural patterns. Perturbations of parent and bifurcating stable POs result in the creation of tori characterized by a smooth color variation on their surface. Furthermore, perturbations of simple unstable parent POs beyond the bifurcation point, which lead to the birth of new stable families of POs, result in the formation of figure-8 structures of smooth color variations. These figure-8 formations surround well-shaped tori around the bifurcated stable POs, losing their well-defined forms for energies further away from the bifurcation point. We also observe that even slight perturbations of highly unstable POs create a cloud of mixed color points, which rapidly move away from the location of the PO. Our study introduces, for the first time, a systematic visualization of 4D surfaces of section within the vicinity of higher multiplicity POs. It elucidates how, in these cases, the coexistence of regular and chaotic orbits contributes to shaping the phase space landscape.

Chaotic dynamics of a three-species food chain model

An analysis of a discrete-time food chain model is presented in this paper to investigate the effects of strong prey pressure on the chain. A bifurcation and stability theory is used to analyze equilibrium formation in system. It is shown by numerical simulations that chaos occurs as a result of the bifurcation of invariant curves following Neimark-Sacker (NS) bifurcation rules. The OGY method on stable periodic orbits of period 1 further controls chaos. Stability is achieved through iterations for chaotic motion controlled by different regulator poles. Numerical simulations are performed to examine the effects of the control method along with the results of our theoretical analysis.

Dynamical analysis of a tabu learning neuron through the discrete implicit mapping method

The complex dynamics of neuron model including bifurcation behaviors and firing patterns, especially unstable firing patterns, furnish a favorable point of view for the research of neurons, which may be of great importance in the prevention of brain-based diseases. To study this issue in depth, the discrete implicit mapping method is employed to evaluate a tabu learning neuron system, which is coupled with electromagnetic excitation and applied driving current in this paper. The bifurcation orbits of periodic motion with rich dynamic phenomena are accurately shown with varying the amplitudes of external input. The evolutions of period-5 to period-20 and period-6 to period-12 can be obtained via period doubling bifurcations and saddle bifurcation. The subcritical pitchfork bifurcation is also exhibited through the complex coexistence behavior of stable and unstable periodic orbits in such a model, which cannot be obtained by the traditional numerical method due to its accumulative errors. The corresponding stabilities and bifurcation of the periodic motions are determined by means of eigenvalues. Therefore, using such a method, the neural spiking events can be developed through phase diagram and time history of membrane potential. Moreover, the tabu learning system is implemented via FPGA (i.e., field programmable gate array), and the unstable and stable phase diagrams are observed completely from oscilloscope, where the experiments results verify the feasibility of semi-analytical solutions. Such investigation will also benefit the development of artificial intelligence and the advances in Brain Medicine.

Bounded orbits for three bodies in ℝ4

We consider the Newtonian 3-body problem in dimension 4, and fix a value of the angular momentum which is compatible with this dimension. We show that the energy function cannot tend to its infimum on an unbounded sequence of states. Consequently the infimum of the energy is its minimum. This completes our previous work [A. Albouy and H. R. Dullin, Relative equilibria of the 3-body problem in [Formula: see text], J. Geom. Mech. 12(3) (2020) 323–341] on the existence of Lyapunov stable relative periodic orbits in the 3-body problem in [Formula: see text].

Multiplicity of Neutrally Stable Periodic Orbits with Coexistence in the Chemostat Subject to Periodic Removal Rate

Multiplicity of Neutrally Stable Periodic Orbits with Coexistence in the Chemostat Subject to Periodic Removal Rate

A Note About the Stable Periodic Orbits for a Class of Three-Dimensional Competitive Systems.

A Note About the Stable Periodic Orbits for a Class of Three-Dimensional Competitive Systems.

Trajectories associated with parabolic and hyperbolic periodic points of piecewise linear area-preserving map

This paper investigates the characteristics of trajectories and invariant sets associated with parabolic and hyperbolic periodic points of a piecewise linear area-preserving map. When the fixed points in both partitions are parabolic, the trajectories may diverge or rotate repeatedly around the origin. By analyzing the trajectory motion in different regions of the phase plane, the characteristics of the trajectory repeated rotation around the origin are shown, and the trajectories may be either divergent or periodic. The necessary conditions for generating stable periodic orbits are presented. It is proved that there are forward invariant sets and that the trajectories from the invariant sets diverge in different quadrants and along a family of invariant lines parallel to the diagonal bisector of the quadrant. We reveal why there are island chains with invariant line segments in a chaotic sea. The method for determining invariant line segments is presented. For the map with two hyperbolic fixed points in both partitions, it is proved that there are forward invariant sets in the phase plane and all trajectories from the invariant sets diverge along a family of hyperbolic invariant sets.

Bifurcation analysis of waning-boosting epidemiological models with repeat infections and varying immunity periods

We consider the SIRWJS epidemiological model that includes the waning and boosting of immunity via secondary infections. We carry out combined analytical and numerical investigations of the dynamics. The formulae describing the existence and stability of equilibria are derived. Combining this analysis with numerical continuation techniques, we construct global bifurcation diagrams with respect to several epidemiological parameters. The bifurcation analysis reveals a very rich structure of possible global dynamics. We show that backward bifurcation is possible at the critical value of the basic reproduction number, R0=1. Furthermore, we find stability switches and Hopf bifurcations from steady states forming multiple endemic bubbles, and saddle–node bifurcations of periodic orbits. Regions of bistability are also found, where either two stable steady states, or a stable steady state and a stable periodic orbit coexist. This work provides an insight to the rich and complicated infectious disease dynamics that can emerge from the waning and boosting of immunity.

Systematically Constructing Mesoscopic Quantum States Relevant to Periodic Orbits in Integrable Billiards from Directionally Resolved Level Distributions

Two-dimensional quantum billiards are one of the most important paradigms for exploring the connection between quantum and classical worlds. Researchers are mainly focused on nonintegrable and irregular shapes to understand the quantum characteristics of chaotic billiards. The emergence of the scarred modes relevant to unstable periodic orbits (POs) is one intriguing finding in nonintegrable quantum billiards. On the other hand, stable POs are abundant in integrable billiards. The quantum wavefunctions associated with stable POs have been shown to play a key role in ballistic transport. A variety of physical systems, such as microwave cavities, optical fibers, optical resonators, vibrating plates, acoustic waves, and liquid surface waves, are used to analogously simulate the wave properties of quantum billiards. This article gives a comprehensive review for the subtle connection between the quantum level clustering and the classical POs for three integrable billiards including square, equilateral triangle, and circular billiards.

Bringing Order to Disorder: A method for stabilising a chaotic system around an arbitrary unstable periodic orbit

This paper introduces a two-step procedure for stabilising a chaotic system around an arbitrary periodic orbit in a Poincaré map. The periodic orbit may be a period-n orbit traversing the Poincaré plane an integer n times before completing the periodic cycle. The order of traversing can also be chosen arbitrarily. Chaotic systems lack predictability due to positive Lyapunov exponents. For this reason, predictive control methods tend to perform poorly because the predictable horizon may be short. Current methods for stabilising chaotic systems work by controlling existing unstable periodic orbits. This paper proposes a method for stabilising the system on an arbitrary point in the Poincaré section. The method in this paper consists of two steps. In the first step, we pose an optimisation problem that computes an input signal that introduces a desired unstable or stable periodic orbit. In the second step, we use existing methods, such as the method using delay coordinate embedding developed by Ott, Grebogi and Yorke in 1990, to stabilise the system around the newly introduced periodic orbit. As an example, we demonstrate the method on the resistively and capacitively shunted driven Josephson junction.

Periodic orbits in a near Yang-Mills potential

We study numerically the orbits in the Yang-Mills (YM) potential and in the potentials of the general form . We found that the stable period-9 periodic orbit of the YM potential belongs to a family of orbits that bifurcates from a basic period-9 family of the general form of the potential V, when α is slightly above zero. This basic period-9 family and its bifurcations exist only up to a maximum value of . We calculate the Hénon stability index of these orbits. The pattern of the stability diagram is the same for all the symmetric orbits of odd periods 3,5,7,9 and 11, that we have found. We also found the stability diagrams for asymmetric orbits of period 2,3,4,5 which have again the same pattern. All these orbits are unstable for α=0 (YM potential) except for the stable orbits of period-9 and some orbits with multiples of 9 periods.

Existence, Number and Stability of Periodic Orbits Induced by Homoclinic Loops in Three-Dimensional Piecewise Linear Systems with an Admissible Saddle-Focus

For a class of three-dimensional piecewise linear systems with an admissible saddle-focus, the existence of three kinds of homoclinic loops is shown. Moreover, the birth and number of the periodic orbits induced by homoclinic bifurcation are investigated, and various sufficient conditions are obtained to guarantee the appearance of only one periodic orbit, finitely many periodic orbits or countably infinitely many periodic orbits. Furthermore, the stability of these newborn periodic orbits is analyzed in detail and some conclusions are made about them to be periodic saddle orbits or periodic sinks. Finally, some examples are given.

Double Hopf bifurcation induced by gestation and memory delays in a spatial model with directed movement

We narrow our focus on double Hopf bifurcations triggered by gestation and memory delays in a spatial model with tactical directed movement in this letter. We generalize the calculation algorithm of normal form concerning the double Hopf bifurcation to a spatial model with different delays in both the reaction and advection terms. We then divide the dynamics around the double Hopf bifurcation points into 6 categories and thus we observe that stable periodic orbits with different spatial modes can coexist.

Minimal model of quasi-cyclic behaviour in turbulence driven by Taylor–Green forcing

We attempt to formulate the simplest possible model mimicking turbulent dynamics, such as quasi-cyclic behaviour (QCB), using only three variables. To this end, we first conduct direct numerical simulations of three-dimensional flow driven by the steady Taylor–Green forcing to find a similarity between a stable periodic orbit (SPO) at a small Reynolds number (Re) and turbulent QCB at higher Re. A close examination of the SPO allows the heuristic formulation of a three-equation model, representing the evolution of Fourier modes in three distinct scales. The model reproduces the continuous bifurcation from SPO to turbulence with QCB when Re is varied. We also demonstrate that, by changing model parameters, the proposed model exhibits a discontinuous transition from steady to chaotic solutions without going through an SPO.