Quasi-periodic Orbits Research Articles

Bifurcations and chaos in two-coupled periodically driven four-well Duffing-van der Pol oscillators

Bifurcations and chaos in two-coupled periodically driven four-well Duffing-van der Pol (DVP) oscillators are numerically investigated as a function of the strength (δ) of nonlinear coupling and the amplitude (f) of the periodic driving force. The influence of weak coupling (δ ≪ 1) and strong coupling (δ ≫ 1) is analyzed with and without the external periodic force. For strong coupling (δ ≫ 1), the system shows completely chaotic behaviour but for weak coupling the system exhibits period-doubling and period-tripling routes to chaos, quasiperiodic orbit, reverse period-doubling bifurcation, periodic windows, chaotic orbit and intermittent behaviours for specific set of values of the parameters. The effect of phase shift (ϕ) of the system is also analyzed. Numerical results are demonstrated by constructing the bifurcation diagram, phase portrait and Poincare´ map.

Detecting unstable periodic orbits and unstable quasiperiodic orbits in vibro-impact systems

In this paper, unstable dynamics is considered for the models of vibro-impact systems with linear differential equations coupled to an impact map. To provide a skeleton for the organization of chaotic attractors, we propose a method for detecting unstable periodic orbits embedded in chaotic attractors through a combination of unconstrained optimization technique and Poincaré map. Three numerical examples from different vibro-impact models demonstrate that the strategy can efficiently detect unstable periodic orbits in chaotic attractors. In order to explore the mechanism responsible for the creation of multi-dimensional tori attractors, we also present another method to detect unstable quasiperiodic orbits embedded multi-dimensional tori attractors by examining a specially transient time series. The upper bound and lower bound of the transient time series (in the Poincaré map) can be obtained by analyzing transient Lyapunov exponent and transient Lyapunov dimension. Some examples verify the effectiveness of the numerical algorithm.

Domains of analyticity and Lindstedt expansions of KAM tori in some dissipative perturbations of Hamiltonian systems

Conformally symplectic systems are characterized by the property that they transform a symplectic form into a multiple of itself. The limit of small dissipation, which is the object of the present study, is particularly interesting.We consider a family of conformally symplectic maps (very simple modifications, which we present, give results for differential equations) defined on a 2d-dimensional symplectic manifold with exact symplectic form ; we assume that satisfies . The d-dimensional parameter μ is called drift. We assume that , where , .We study the perturbative expansions and the domains of analyticity in near of the parameterization of the quasi-periodic orbits of frequency ω (assumed to be Diophantine) and of the parameter μ. Notice that this is a singular perturbation, since any friction (no matter how small) reduces the set of quasi-periodic solutions in the system. We prove that the tori are analytic in a domain in the complex plane, obtained by taking from a ball centered at zero a sequence of smaller balls with center along smooth lines going through the origin. The radii of the excluded balls decrease faster than any power of the distance of the center to the origin. We state also a conjecture on the optimality of our results. The boundary of the domain is very thin, so that we can perform unique analytic continuation of the invariant tori along closed circles enclosing the points in the complement of the analyticity. We show that there is no monodromy of these continuations, either for the tori, for the invariant manifolds or for the drift.The proof is based on the following procedure. To find a quasi-periodic solution, one solves an invariance equation for the embedding of the torus, depending on the parameters of the family. Assuming that the frequency of the torus satisfies a Diophantine condition, under mild non-degeneracy assumptions, using a Lindstedt procedure we construct an approximate solution to all orders of the invariance equation describing the KAM torus. Starting from such approximate solution, we use an a posteriori KAM theorem to get the true solution of the invariance equation. This allows also the study of monogenic and Whitney differentiability properties of the extensions as well as the monodromy.

Discontinuity Induced Hopf and Neimark–Sacker Bifurcations in a Memristive Murali–Lakshmanan–Chua Circuit

We report using Clarke’s concept of generalized differential and a modification of Floquet theory to nonsmooth oscillations, the occurrence of discontinuity induced Hopf bifurcations and Neimark–Sacker bifurcations leading to quasiperiodic attractors in a memristive Murali–Lakshmanan–Chua (memristive MLC) circuit. The above bifurcations arise because of the fact that a memristive MLC circuit is basically a nonsmooth system by virtue of having a memristive element as its nonlinearity. The switching and modulating properties of the memristor which we have considered endow the circuit with two discontinuity boundaries and multiple equilibrium points as well. As the Jacobian matrices about these equilibrium points are noninvertible, they are nonhyperbolic, some of these admit local bifurcations as well. Consequently when these equilibrium points are perturbed, they lose their stability giving rise to quasiperiodic orbits. The numerical simulations carried out by incorporating proper discontinuity mappings (DMs), such as the Poincaré discontinuity map (PDM) and zero time discontinuity map (ZDM), are found to agree well with experimental observations.

Formation flying on quasi-halo orbits in restricted Sun–Earth/Moon system

Due to the strong nonlinear perturbations near the libration points, the continuous low-thrust technique has many potential applications in stationkeeping relative motions. The Hamiltonian structure-preserving (HSP) control is employed in this paper to stabilize formation flying on quasi-periodic orbits near LL1 of the restricted Sun–Earth–Moon–spacecraft system. In the bi-circular model (BCM), a multiple shooting corrector is developed to refine quasi-periodic orbits as the chief spacecraft's reference trajectories. The linearized variation equation in BCM is used to design the stationkeeping control. A HSP controller is constructed to change the topology of the equilibrium from hyperbolic to elliptic using only relative position feedbacks consisting of stable, unstable and center manifolds. The critical control gains for transient and long-term stabilities are presented to guide the selection of control gains.

BIFURCATION AND CHAOS IN A DISCRETE TIME PREDATOR-PREY SYSTEM OF LESLIE TYPE WITH GENERALIZED HOLLING TYPE Ⅲ FUNCTIONAL RESPONSE

This paper is devoted to study a discrete time predator-prey system of Leslie type with generalized Holling type Ⅲ functional response obtained using the forward Euler scheme. Taking the integration step size as the bifurcation parameter and using the center manifold theory and bifurcation theory, it is shown that by varying the parameter the system undergoes flip bifurcation and Neimark-Sacker bifurcation in the interior of R₊². Numerical simulations are implemented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as cascade of period-doubling bifurcation in period-2, 4, 8, quasi-periodic orbits and the chaotic sets. These results shows much richer dynamics of the discrete model compared with the continuous model. The maximum Lyapunov exponent is numerically computed to confirm the complexity of the dynamical behaviors. Moreover, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.

Bifurcation analysis of the three-dimensional Hénon map

In this paper, we consider the dynamics of a generalized three-dimensional Henon map. Necessary and sufficient conditions on the existence and stability of the fixed points of this system are established. By applying the center manifold theorem and bifurcation theory, we show that the system has the fold bifurcation, flip bifurcation, and Neimark-Sacker bifurcation under certain conditions. Numerical simulations are presented to not only show the consistence between examples and our theoretical analysis, but also exhibit complexity and interesting dynamical behaviors, including period-10, -13, -14, -16, -17, -20, and -34 orbits, quasi-periodic orbits, chaotic behaviors which appear and disappear suddenly, coexisting chaotic attractors. These results demonstrate relatively rich dynamical behaviors of the three-dimensional Henon map.

COMPLEX DYNAMIC BEHAVIORS OF A DISCRETE-TIME PREDATOR-PREY SYSTEM

The dynamics of a discrete-time predator-prey system is investigated in detail in this paper. It is shown that the system undergoes flip bifurcation and Hopf bifurcation by using center manifold theorem and bifurcation theory. Furthermore, Marotto's chaos is proved when some certain conditions are satisfied. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-6, 7, 8, 10, 14, 18, 24, 36, 50 orbits, attracting invariant cycles, quasi-periodic orbits, nice chaotic behaviors which appear and disappear suddenly, coexisting chaotic attractors, etc. These results reveal far richer dynamics of the discrete-time predator-prey system. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.

Dynamics of Miura-patterned foldable sheets in shear flow.

We study the dynamics of piecewise rigid sheets containing predefined crease lines in shear flow. The crease lines act like hinge joints along which the sheet may fold rigidly, i.e. without bending any other crease line. We choose the crease lines such that they tessellate the sheet into a two-dimensional array of parallelograms. Specifically, we focus on a particular arrangement of crease lines known as a Miura-pattern in the origami community. When all the hinges are fully open the sheet is planar, whereas when all are closed the sheet folds over itself to form a compact flat structure. Due to rigidity constraints, the folded state of a Miura-sheet can be described using a single fold angle. The hinged sheet is modeled using the framework of constrained multibody systems in the absence of inertia. The hydrodynamic drag on each of the rigid panels is calculated based on an inscribed elliptic disk, but intra-panel hydrodynamic interactions are neglected. We find that when the motion of a sheet remains symmetric with respect to the flow-gradient plane, after a sufficiently long time, the sheet either exhibits asymptotically periodic tumbling and breathing, indicating approach to a limit cycle; or it reaches a steady state by completely unfolding, which we show to be a half-stable node in the phase space. In the case of asymmetric motion of the sheet with respect to the flow-gradient plane, we find that the terminal state of motion is one of - (i) steady state with a fully unfolded or fully folded configuration, (ii) asymptotically periodic tumbling, indicating approach to a limit cycle, (iii) cyclic tumbling without repetition, indicating a quasiperiodic orbit, or (iv) cyclic tumbling with repetition after several cycles, indicating a resonant quasiperiodic orbit. No chaotic behavior was found.

Complex dynamic behaviour of an economic cycle model

In this paper, we investigate the dynamics of a nonlinear economic cycle model. The necessary and sufficient conditions are given to guarantee the existence and stability of the fixed point. It is also shown that the system undergoes a Neimark–Sacker bifurcation by using center manifold theorem and bifurcation theory. Furthermore, Marotto’s chaos is proved when certain conditions are satisfied. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviour, such as the period-10, -16, -20 orbits, attracting invariant cycles, quasi-periodic orbits, 10-coexisting chaotic attractors, and boundary crisis. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.

The theory of asynchronous relative motion I: time transformations and nonlinear corrections

Using alternative independent variables in lieu of time has important advantages when propagating the partial derivatives of the trajectory. This paper focuses on spacecraft relative motion, but the concepts presented here can be extended to any problem involving the variational equations of orbital motion. A usual approach for modeling the relative dynamics is to evaluate how the reference orbit changes when modifying the initial conditions slightly. But when the time is a mere dependent variable, changes in the initial conditions will result in changes in time as well: a time delay between the reference and the neighbor solution will appear. The theory of asynchronous relative motion shows how the time delay can be corrected to recover the physical sense of the solution and, more importantly, how this correction can be used to improve significantly the accuracy of the linear solutions to relative motion found in the literature. As an example, an improved version of the Clohessy-Wiltshire (CW) solution is presented explicitly. The correcting terms are extremely compact, and the solution proves more accurate than the second and even third order CW equations for long propagations. The application to the elliptic case is also discussed. The theory is not restricted to Keplerian orbits, as it holds under any perturbation. To prove this statement, two examples of realistic trajectories are presented: a pair of spacecraft orbiting the Earth and perturbed by a realistic force model; and two probes describing a quasi-periodic orbit in the Jupiter-Europa system subject to third-body perturbations. The numerical examples show that the new theory yields reductions in the propagation error of several orders of magnitude, both in position and velocity, when compared to the linear approach.

De Sitter’s theory of Galilean satellites

In this article, we investigate the mathematical part of De Sitter’s theory on the Galilean satellites, and further extend this theory by showing the existence of some quasi-periodic librating orbits by application of KAM theorems. After showing the existence of De Sitter’s family of linearly stable periodic orbits in the Jupiter–Io–Europa–Ganymede model by averaging and reduction techniques in the Hamiltonian framework, we further discuss the possible extension of this theory to include a fourth satellite Callisto, and establish the existence of a set of positive measure of quasi-periodic librating orbits in both models for almost all choices of masses among which one sufficiently dominates the others.

Application of Integral Criterion for the Research of Periodic and Quasi-Periodic Orbits in the Three-Body Problem

Рассматривается возможность и целесообразность применения интегрального критерия, осно- ванного на принципе наименьшего действия, для определения области существования перио- дических по Пуанкаре и квазипериодических орбит в задаче трех тел. Показано, что, несмотря на наличие нескольких типов критериев существования периодических орбит, поиск рациональных вариантов вычислительных процедур соответствующего вида продолжает оставаться актуальной задачей. Описываемый вариант решения позволяет выделить области возможного существования периодических орбит задачи трех тел, а также осуществить с высокой надежностью кластеризацию этих орбит по различным параметрам, таким как направление обращения, средний период обращения, средний эксцентриситет, среднее изменение углового положения линии апсид и другим.

Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction

We propose a unified approach to nonlinear modal analysis in dissipative oscillatory systems. This approach eliminates conflicting definitions, covers both autonomous and time-dependent systems and provides exact mathematical existence, uniqueness and robustness results. In this setting, a nonlinear normal mode (NNM) is a set filled with small-amplitude recurrent motions: a fixed point, a periodic orbit or the closure of a quasiperiodic orbit. In contrast, a spectral submanifold (SSM) is an invariant manifold asymptotic to a NNM, serving as the smoothest nonlinear continuation of a spectral subspace of the linearized system along the NNM. The existence and uniqueness of SSMs turns out to depend on a spectral quotient computed from the real part of the spectrum of the linearized system. This quotient may well be large even for small dissipation; thus, the inclusion of damping is essential for firm conclusions about NNMs, SSMs and the reduced-order models they yield.

Stability of the celestial body orbits under the influence of Yukawa potential

The famous Yukawa-type potential was very often used for explaining a great variety of recent observed astronomical phenomena which range from the solar-system scale to cosmological distances. In this paper we tackle the two-body problem in the Yukawa post-Newtonian field from the particular standpoint of orbits stability. Starting from the equations of motion and first integrals written in standard polar coordinates, we apply McGehee-type transformations of the second kind. Then we depict the phase-space structure considering the foliations by the energy constant and the angular momentum constant. Various stability regions are found for each case. The problem presents interesting features, such as: cases when all trajectories (maybe except for a separatrix) are stable; cases when there exist totally different types of orbits, both stable and unstable, for the same values of the energy constant and the angular momentum constant; the existence of stable motion for nonnegative energy levels; positive Lebesgue measure for initial data leading to quasiperiodic and noncircular periodic orbits; the key role of the angular momentum.

Codimension-Two Bifurcation, Chaos and Control in a Discrete-Time Information Diffusion Model

In this paper, we present a discrete model to illustrate how two pieces of information interact with online social networks and investigate the dynamics of discrete-time information diffusion model in three types: reverse type, intervention type and mutualistic type. It is found that the model has orbits with period 2, 4, 6, 8, 12, 16, 20, 30, quasiperiodic orbit, and undergoes heteroclinic bifurcation near 1:2 point, a homoclinic structure near 1:3 resonance point and an invariant cycle bifurcated by period 4 orbit near 1:4 resonance point. Moreover, in order to regulate information diffusion process and information security, we give two control strategies, the hybrid control method and the feedback controller of polynomial functions, to control chaos, flip bifurcation, 1:2, 1:3 and 1:4 resonances, respectively, in the two-dimensional discrete system.

Dynamics of a discrete-time predator-prey system

We investigate the dynamics of a discrete-time predator-prey system. Firstly, we give necessary and sufficient conditions of the existence and stability of the fixed points. Secondly, we show that the system undergoes a flip bifurcation and a Neimark-Sacker bifurcation by using center manifold theorem and bifurcation theory. Furthermore, we present numerical simulations not only to show the consistence with our theoretical analysis, but also to exhibit the complex but interesting dynamical behaviors, such as the period-6, -11, -16, -18, -20, -21, -24, -27, and -37 orbits, attracting invariant cycles, quasi-periodic orbits, nice chaotic behaviors, which appear and disappear suddenly, coexisting chaotic attractors, etc. These results reveal far richer dynamics of the discrete-time predator-prey system. Finally, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.

Nonlinear dynamics of a generalized higher-order nonlinear Schrödinger equation with a periodic external perturbation

The nonlinear dynamics of a generalized higher-order nonlinear Schrodinger (HNLS) equation with a periodic external perturbation is investigated numerically. Via the phase plane analysis, we find that both the homoclinic orbits and heteroclinic orbits can exist for the unperturbed HNLS equation under certain conditions, which respectively corresponds to the bell-shaped and kink-shaped soliton solutions. Moreover, under the effect of the periodic external perturbation, the quasi-periodic bifurcations arise and can evolve into the chaos. The dynamical responses of the perturbed system varying with the perturbation strength and two types of chaotic attractors are discussed to show the existence of the chaotic motions. Via the feedback control methods, such chaotic motions are found to be controlled effectively and finally evolve into the stable quasi-periodic orbits. All the results are helpful to understand the dynamical properties of the nonlinear system.

Rapid trajectory design in the Earth–Moon ephemeris system via an interactive catalog of periodic and quasi-periodic orbits

Upcoming missions and prospective design concepts in the Earth–Moon system extensively leverage multi-body dynamics that may facilitate access to strategic locations or reduce propellant usage. To incorporate these dynamical structures into the mission design process, Purdue University and the NASA Goddard Flight Space Center have initiated the construction of a trajectory design framework to rapidly access and compare solutions from the circular restricted three-body problem. This framework, based upon a ‘dynamic’ catalog of periodic and quasi-periodic orbits within the Earth–Moon system, can guide an end-to-end trajectory design in an ephemeris model. In particular, the inclusion of quasi-periodic orbits further expands the design space, potentially enabling the detection of additional orbit options. To demonstrate the concept of a ‘dynamic’ catalog, a prototype graphical interface is developed. Strategies to characterize and represent periodic and quasi-periodic information for interactive trajectory comparison and selection are discussed. Two sample applications for formation flying near the Earth–Moon L2 point and lunar space infrastructures are explored to demonstrate the efficacy of a ‘dynamic’ catalog for rapid trajectory design and validity in higher-fidelity models.

Dynamic analysis of discrete-time, continuous-time and delayed feedback jerky equations

This paper is devoted to the dynamics of a discrete-time and continuous-time jerk class \({ JD }_{1}\) which exhibit multiple complex patterns as parameters vary. It is shown that the discrete version of jerk class \({ JD }_{1}\) undergoes Neimark–Sacker bifurcation by applying center manifold theorem and normal form theory. Numerical simulations are given not only to support theoretical analyses but also to show complex dynamical behaviors, including a family of closed invariant curves, periodic-12, 47 orbits, quasi-periodic orbit and chaotic oscillations. For the continuous version jerky class \({ JD }_{1}\), the direction of Hopf bifurcation is derived, and the expression of bifurcating periodic solution is presented. Besides, constant control, state feedback control and time-delayed feedback control are considered in the continuous jerk class.