Stable Periodic Orbits Research Articles

Augmented phase reduction for periodic orbits near a homoclinic bifurcation and for relaxation oscillators

Oscillators - dynamical systems with stable periodic orbits - arise in many systems of physical, technological, and biological interest. The standard phase reduction, a model reduction technique based on isochrons, can be unsuitable for oscillators which have a small-magnitude negative nontrivial Floquet exponent. This necessitates the use of the augmented phase reduction, a recently devised two-dimensional reduction technique based on isochrons and isostables. In this article, we calculate analytical expressions for the augmented phase reduction for two dynamically different planar systems: periodic orbits born out of a homoclinic bifurcation, and relaxation oscillators. To validate our calculations, we simulate models in these dynamic regimes, and compare their numerically computed augmented phase reduction with the derived analytical expressions. These analytical and numerical calculations help us to understand conditions for which the use of augmented phase reduction over the standard phase reduction can be advantageous.

Mathematical Analysis of a Three-Tiered Model of Anaerobic Digestion

In this paper, we are interested in a mechanistic model describing the anaerobic mineralization of chlorophenol in a three-step food-web. The model is a six-dimensional system of ordinary differential equations. In our study, we take into account the phenol and the hydrogen input concentrations as well as the maintenance terms. Moreover, we consider the case of a large class of growth rates, instead of specific kinetics. In this general case, a recent study shows that the system can have up to eight steady states and their existence conditions were analytically determined. We focus here on the necessary and sufficient conditions of the local stability of the steady states, according to the four operating parameters of the process, which are the dilution rate and the chlorophenol, phenol and hydrogen input concentrations. In previous studies, this stability analysis was performed only numerically. Using the Lienard-Chipart stability criterion, we show that the positive steady state can be unstable and we give numerical evidence for a supercritical Hopf bifurcation with the appearance of a stable periodic orbit. We give two bifurcation diagrams with respect to the dilution rate, first, and then to the chlorophenol input concentration as the bifurcating parameters, showing that the system can present rich behavior including bistability, coexistence and occurrence of limit cycle.

Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays

<p style='text-indent:20px;'>We consider a two-species Lotka-Volterra competition system with both local and nonlocal intraspecific and interspecific competitions under the homogeneous Neumann condition. Firstly, we obtain conditions for the existence of Hopf, Turing, Turing-Hopf bifurcations and the necessary and sufficient condition that Turing instability occurs in the weak competition case, and find that the strength of nonlocal intraspecific competitions is the key factor for the stability of coexistence equilibrium. Secondly, we derive explicit formulas of normal forms up to order 3 by applying center manifold theory and normal form method, in which we show the difference compared with system without nonlocal terms in calculating coefficients of normal forms. Thirdly, the existence of complex spatiotemporal phenomena, such as the spatial homogeneous periodic orbit, a pair of stable spatial inhomogeneous steady states and a pair of stable spatial inhomogeneous periodic orbits, is rigorously proved by analyzing the amplitude equations. It is shown that suitably strong nonlocal intraspecific competitions and nonlocal delays can result in various coexistence states for the competition system in the weak competition case. Lastly, these complex spatiotemporal patterns are presented in the numerical results.</p>

On the design of stable periodic orbits of a triple pendulum on a cart with experimental validation

In this paper, we enhance an inversion-based control approach towards the stabilization of a periodic orbit of a multi-link triple pendulum on a cart. To this end, a nominal trajectory is obtained by formulating the considered transition problem as a two-point boundary value problem (BVP) in input–output representation. For solvability of the resulting BVP, a setup function is introduced such that additional parameters are provided in the differential equation of the internal dynamics. Based on the linearized dynamics about the nominal trajectory, a linear-quadratic-Gaussian (LQG) controller is implemented to compensate for measurement noise, model uncertainties, and external disturbances. This way we achieve to force a triple pendulum to move along a non-trivial periodic orbit and render it attractive. The high performance and accuracy of our approach is illustrated on an experimental setup.

A Predator-Prey model in the chemostat with Holling Type II response function

A model of predator-prey interaction in a chemostat with Holling Type II functional and numerical response functions of the Monod or Michaelis-Menten form is considered. It is proved that local asymptotic stability of the coexistence equilibrium implies that it is globally asymptotically stable. It is also shown that when the coexistence equilibrium exists but is unstable, solutions converge to a unique, orbitally asymptotically stable periodic orbit. Thus the range of the dynamics of the chemostat predator-prey model is the same as for the analogous classical Rosenzweig-MacArthur predator-prey model with Holling Type II functional response. An extension that applies to other functional rsponses is also given.

One family of 13315 stable periodic orbits of non-hierarchical unequal-mass triple systems

The three-body problem has been studied for more than three centuries [1,2], and has received much attention in recent years [3-5]. It shows complex dynamical phenomena due to the mutual gravitational interaction of the three bodies. Triple systems are common in astronomy, but all observed periodic triple systems are hierarchical up till now [6-8]. It is traditionally believed that bound non-hierarchical triple systems are almost unstable and disintegrate into a stable binary system and a single star [5], and thus stable periodic orbits of non-hierarchical triple systems are rather scarce. Here we report one family of 13315 stable periodic orbits of the non-hierarchical triple system with unequal mass. Compared with the narrow mass region (only 10E-5) of the stable figure-eight solution [9], our newly-found stable periodic orbits can have fairly large mass region. It is found that many of these newly-found stable periodic orbits have the mass ratios close to those of the hierarchical triple systems that have been measured by the astronomical observation. It implies that these stable periodic orbits of the non-hierarchical triple system with distinctly unequal masses can be quite possibly observed in practice. Our investigation also suggests that there should exist an infinite number of stable periodic orbits of non-hierarchical triple systems with distinctly unequal masses. Obviously, these stable periodic orbits of the non-hierarchical unequal-mass triple system have broad impact for the astrophysical scenario: they could inspire the theoretical and observational study of the non-hierarchical triple system, the formation of triple stars [6], the gravitational waves pattern [10] and the gravitational waves observation [11] of the non-hierarchical triple system.

Jacobi analysis of a segmented disc dynamo system

In this paper, Jacobi stability of a segmented disc dynamo system is geometrically investigated from viewpoint of Kosambi–Cartan–Chern (KCC) theory in Finsler geometry. First, the geometric objects associated to the reformulated system are explicitly obtained. Second, the Jacobi stability of equilibria and a periodic orbit are discussed in the light of deviation curvature tensor. It is shown that all the equilibria are always Jacobi unstable for any parameters, a Lyapunov stable periodic orbit falls into both Jacobi stable regions and Jacobi unstable regions. The considered system is not robust to small perturbations of the equilibria, and some fragments of the periodic orbit are included in fragile region, indicating that the system is extremely sensitive to internal parameters and environment. Finally, the dynamics of the deviation vector and its curvature near all the equilibria are presented to interpret the onset of chaos in the dynamo system. In a physical sense, magnetic fluxes and angular velocity can show irregular oscillations under some certain cases, these oscillations may reveal the irregularity of magnetic field’s evolution and reversals.

Primary and subharmonic simultaneous resonance of fractional-order Duffing oscillator

The primary and subharmonic simultaneous resonance of Duffing oscillator with fractional-order derivative is studied. Firstly, the approximately analytical solution of the resonance is obtained by the method of multiple scales, and the correctness and satisfactory precision of the analytical solution are verified by numerical simulation. Then, the amplitude–frequency curve equation and phase–frequency curve equation are derived from the analytical solution. The stability condition of the steady-state response is obtained by Lyapunov’s first method, and the state switching between two stable periodic orbits is demonstrated. Finally, the effects of nonlinear factor on the system response are analyzed, and the difference between stiffness softening and stiffness hardening system is demonstrated. The influence of fractional-order term on the system is analyzed in depth, and the effect mechanism of fractional-order term is revealed, i.e., the focus and intensity of effect are determined by the order and coefficient of the fractional-order derivative, respectively.

Rich dynamics of a simple delay host-pathogen model of cell-to-cell infection for plant virus

Viral dynamics within plant hosts can be important for understanding plant disease prevalence and impacts. However, few mathematical modeling efforts aim to characterize within-plant viral dynamics. In this paper, we derive a simple system of delay differential equations that describes the spread of infection throughout the plant by barley and cereal yellow dwarf viruses via the cell-to-cell mechanism. By incorporating ratio-dependent incidence function and logistic growth of the healthy cells, the model can capture a wide range of biologically relevant phenomena via the disease-free, endemic, mutual extinction steady states, and a stable periodic orbit. We show that when the basic reproduction number is less than \begin{document}$ 1 $\end{document} ( \begin{document}$ R_0 ), the disease-free steady state is asymptotically stable. When \begin{document}$ R_0>1 $\end{document} , the dynamics either converge to the endemic equilibrium or enter a periodic orbit. Using a ratio-dependent transformation, we show that if the infection rate is very high relative to the growth rate of healthy cells, then the system collapses to the mutual extinction steady state. Numerical and bifurcation simulations are provided to demonstrate our theoretical results. Finally, we carry out parameter estimation using experimental data to characterize the effects of varying nutrients on the dynamics of the system. Our parameter estimates suggest that varying the nutrient supply of nitrogen and phosphorous can alter the dynamics of the infection in plants, specifically reducing the rate of viral production and the rate of infection in certain cases.

Stability of forced-damped response in mechanical systems from a Melnikov analysis.

Frequency responses of multi-degree-of-freedom mechanical systems with weak forcing and damping can be studied as perturbations from their conservative limit. Specifically, recent results show how bifurcations near resonances can be predicted analytically from conservative families of periodic orbits (nonlinear normal modes). However, the stability of forced-damped motions is generally determined a posteriori via numerical simulations. In this paper, we present analytic results on the stability of periodic orbits that perturb from conservative nonlinear normal modes. In contrast with prior approaches to the same problem, our method can tackle strongly nonlinear oscillations, high-order resonances, and arbitrary types of non-conservative forces affecting the system, as we show with specific examples.

Exploiting periodic orbits as dynamical clues for Kepler and K2 systems

Aims. Many extrasolar systems possessing planets in mean-motion resonance or resonant chain have been discovered to date. The transit method coupled with transit timing variation analysis provides an insight into the physical and orbital parameters of the systems, but suffers from observational limitations. When a (near-)resonant planetary system resides in the dynamical neighbourhood of a stable periodic orbit, its long-term stability, and thus survival, can be guaranteed. We use the intrinsic property of the periodic orbits, namely their linear horizontal and vertical stability, to validate or further constrain the orbital elements of detected two-planet systems. Methods. We computed the families of periodic orbits in the general three-body problem for several two-planet Kepler and K2 systems. The dynamical neighbourhood of the systems is unveiled with maps of dynamical stability. Results. Additional validations or constraints on the orbital elements of K2-21, K2-24, Kepler-9, and (non-coplanar) Kepler-108 near-resonant systems were achieved. While a mean-motion resonance locking protects the long-term evolution of the systems K2-21 and K2-24, such a resonant evolution is not possible for the Kepler-9 system, whose stability is maintained through an apsidal anti-alignment. For the Kepler-108 system, we find that the stability of its mutually inclined planets could be justified either solely by a mean-motion resonance, or in tandem with an inclination-type resonance. Going forward, dynamical analyses based on periodic orbits could yield better constrained orbital elements of near-resonant extrasolar systems when performed in parallel to the fitting of the observational data.

Jeffery orbits for an object with discrete rotational symmetry

We theoretically investigate the motions of an object immersed in a background flow at a low Reynolds number, generalizing the Jeffery equation for the angular dynamics to the case of an object with n-fold rotational symmetry (n ≥ 3). We demonstrate that when n ≥ 4, the dynamics are identical to those of a helicoidal object for which two parameters related to the shape of the object, namely, the Bretherton constant and a chirality parameter, determine the dynamics. When n = 3, however, we find that the equations require a new parameter that is related to the shape and represents the strength of triangularity. On the basis of detailed symmetry arguments, we show theoretically that microscopic objects can be categorized into a small number of classes that exhibit different dynamics in a background flow. We perform further analyses of the angular dynamics in a simple shear flow, and we find that the presence of triangularity can lead to chaotic angular dynamics, although the dynamics typically possess stable periodic orbits, as further demonstrated by an example of a triangular object. Our findings provide a comprehensive viewpoint concerning the dynamics of an object in a flow, emphasizing the notable simplification of the dynamics resulting from the symmetry of the object’s shape, and they will be useful in studies of fluid–structure interactions at a low Reynolds number.

Kantowski-Sachs Einstein-aether scalar field cosmological models: the sequel

Utilizing the autonomous system of ordinary differential equations derived in [1] to define the evolution, we further investigate a class of cosmological models within an Einstein-aether gravitational framework by introducing a non-trivial coupling between the shear of the aether field to the scalar field on the future asymptotic solution. We subsequently conduct qualitative and numerical stability analysis on the new set of equilibrium points and paramountly determine that the expansionary power-law inflationary attractor becomes anisotropic rather than isotropic in the presence of such a coupling. It is further shown that the stability of this solution is dependent on the value of the shear coupling parameter a3. We also discover a family of asymptotically stable periodic orbits which exist for a particular range of parameter values within the Bianchi I invariant set and vanish in the absence of coupling between the aether field and the scalar field.

Improving empirical characteristic multiplier estimation through a change of basis

This paper applies empirical Floquet theory to extract characteristic multipliers from time series data and builds upon prior works by significantly reducing the influence of experimental noise. Characteristic multipliers (CMs), which are the eigenvalues of the transition matrix governing the evolution of transient solutions over one period of motion, quantify the local stability of periodic orbits and can be estimated experimentally without knowledge of the system model. Traditionally, empirical CM estimates were obtained from the transient dynamics observed by subtracting measured steady-state behavior from a recorded perturbed response. The subtraction of two experimentally measured quantities amplifies noise, producing inaccurate CM estimates. By applying a moving integral to isolate the transient dynamics, this work provides an approach to reduce, rather than amplify, the influence of noise on empirical CM results. This approach is applied to the reconstructed phase space of a numerical example and an experimental system to demonstrate the improvements.

On the Behavior of the Generalized Alignment Index (GALI) Method for Regular Motion in Multidimensional Hamiltonian Systems

We investigate the behavior of the Generalized Alignment Index of order k (GALIk ) for regular orbits of multidimensional Hamiltonian systems. The GALIk is an efficient chaos indicator, which asymptotically attains positive values for regular motion when 2≤k ≤N, with N being the dimension (D) of the torus on which the motion occurs. By considering several regular orbits in the neighborhood of two typical simple, stable periodic orbits of the Fermi-Pasta-Ulam-Tsingou (FPUT) β model for various values of the system's degrees of freedom, we show that the asymptotic GALIk values decrease when the order k of the index increases and when the orbit's energy approaches the periodic orbit's destabilization energy where the stability island vanishes, while they increase when the considered regular orbit moves further away from the periodic one for a fixed energy. In addition, by performing extensive numerical simulations we show that the behavior of the index does not depend on the choice of the initial deviation vectors needed for its evaluation.

Stable and unstable periodic orbits and their bifurcations in the nonlinear dynamical system with a fixed point vortex in a periodic flow

In this paper, periodic orbits in the nonlinear dynamical system with a fixed point vortex in a periodic flow are investigated. Under the influence of periodic perturbations in the phase space, an infinite number of nonlinear resonances with elliptic and hyperbolic periodic orbits arise. It is shown that these orbits exist even with completely destroyed resonant islands. In the perturbed system, all periodic orbits with periods up to T=4T0, where T0 is period of perturbation, are found. The existence of nonlinear resonances of the KAM and non-KAM nature is shown, and a genetic relationship between different orbits is established. All elliptic orbits are destroyed with an increase of perturbation by the universal cascade of period doubling. It is shown that the rates of cascades for different orbits have close values and are consistent with the value of the Feigenbaum constant for two-dimensional conservative mappings. The complex interaction of hyperbolic orbits of the secondary resonances with the elliptic orbit of the primary resonance is demonstrated. It is shown that in addition to the universal cascade of period doubling, other bifurcation scenarios common to different orbits are possible.

New insights into a chaotic system with only a Lyapunov stable equilibrium

This paper gives some new insights into a chaotic system. The considered system has only one Lyapunov stable equilibrium and positive Lyapunov exponent in some certain parameter range, which means there coexists chaotic attractors and Lyapunov stable equibirium in system. First, from the perspective of analyzing the global structure of the system, based on the Poincaré compactification technique, the complete description of its dynamical behavior on the sphere at infinity is presented. The obtaining results show that its global dynamical behavior is very complex. Second, from a geometric perspective, the Jacobi stability of the system is investigated, including the unique equilibrium and a periodic orbit. Interestingly, we find that the unique Lyapunov stable equilibrium is always Jacobi unstable, a Lyapunov stable periodic orbit falls into both Jacobi stable and Jacobi unstable regions. It is shown that we might witness chaotic behavior of the system before the trajectories enter a neighborhood of the equilibrium point. It is hoped that the investigation of this paper can contribute to better understand and study the complex chaotic system with stable equilibria.

Global Dynamics and Bifurcation of Periodic Orbits in a Modified Nosé-Hoover Oscillator

We perform a global dynamical analysis of a modified Nosé-Hoover oscillator, obtained as the perturbation of an integrable differential system. Using this new approach for studying such an oscillator, in the integrable cases, we give a complete description of the solutions in the phase space, including the dynamics at infinity via the Poincaré compactification. Then using the averaging theory, we prove analytically the existence of a linearly stable periodic orbit which bifurcates from one of the infinite periodic orbits which exist in the integrable cases. Moreover, by a detailed numerical study, we show the existence of nested invariant tori around the bifurcating periodic orbit. Finally, starting with the integrable cases and increasing the parameter values, we show that chaotic dynamics may occur, due to the break of such an invariant tori, leading to the creation of chaotic seas surrounding regular regions in the phase space.

Retrograde periodic orbits in 1/2, 2/3 and 3/4 mean motion resonances with Neptune

We study planar and three-dimensional retrograde periodic orbits, using the model of the restricted three-body problem (RTBP) with the Sun and Neptune as primaries and focusing on the dynamics of resonant trans-Neptunian objects (TNOs). The position and the stability character of the periodic orbits can provide important piece of information on the stability and long-term evolution of small TNOs in retrograde motion. Using the circular planar model as the basic model, families of retrograde symmetric periodic orbits are computed at the 1/2, 2/3 and 3/4 exterior mean motion resonances with Neptune. The bifurcations for planar families of the elliptic model and families of the circular spatial model are determined and the bifurcated families are computed. In our study of the planar elliptic model, we consider the eccentricity of the primaries in the whole interval $$0<e'<1$$ for dynamical completeness. In the spatial circular model, retrograde periodic orbits are obtained mainly from bifurcations of the retrograde planar orbits. Also, we obtain retrograde periodic motion from continuing direct orbits for inclination values larger than $$90^\circ $$ . The linear stability of orbits is of major importance. Generally, stable periodic orbits are associated with phase space domains of resonant motion where TNOs can be captured. TNOs of retrograde motion are not common, but new discoveries cannot be excluded.

Trapping of swimmers in a vortex lattice.

We examine the motion of rigid, ellipsoidal swimmers subjected to a steady vortex flow in two dimensions. Numerical simulations of swimmers in a spatially periodic array of vortices reveal a range of possible behaviors, including trapping inside a single vortex and motility-induced diffusion across many vortices. While the trapping probability vanishes at a sufficiently high swimming speed, we find that it exhibits surprisingly large oscillations as this critical swimming speed is approached. Strikingly, at even higher swimming speeds, we find swimmers that swim perpendicular to their elongation direction can again become trapped. To explain this complex behavior, we investigate the underlying swimmer phase-space geometry. We identify the fixed points and periodic orbits of the swimmer equations of motion that regulate swimmer trapping inside a single vortex cell. For low to intermediate swimming speeds, we find that a stable periodic orbit surrounded by invariant tori forms a transport barrier to swimmers and can trap them inside individual vortices. For swimming speeds approaching the maximum fluid speed, we find instead that perpendicular swimmers can be trapped by asymptotically stable fixed points. A bifurcation analysis of the stable periodic orbit and the fixed points explains the complex and non-monotonic breakdown and re-emergence of swimmer trapping as the swimmer speed and shape are varied.