Higher Periodic Orbits Research Articles

The Periodic Integer Orbits of Polynomial Recursions With Integer Coefficients

We show that polynomial recursions $x_{n+1}=x_{n}^{m}-k$ where $k,m$ are integers and $m$ is positive have no nontrivial periodic integral orbits for $m\geq3$. If $m=2$ then the recursion has integral two-cycles for infinitely many values of $k$ but no higher period orbits. We also show that these statements are true for all quadratic recursions.

Orbital Stability and Invariant Manifolds on Distant Retrograde Orbits around Ganymede and Nearby Higher-Period Orbits

In the past few years, distant retrograde orbits (DROs) have become increasingly popular due to their conspicuous stability. Nevertheless, it is this characteristic that results in the challenge to the design of transfer orbits into/out of DROs. This paper investigates the DROs around Ganymede in order to utilize their dynamical characteristics for Jupiter system exploration. In particular, the DRO family is calculated by numerical integration and numerical continuation, higher-period orbits near the DROs are detected using bifurcation theory, and characteristics including orbital stability and invariant manifolds of these orbits are investigated through stability indices and manifold theory. The stability of DROs and the higher-period orbits are first investigated in the circular restricted three-body problem and are then verified in a third-body gravitation perturbation model. The results show that the strong stability of DROs makes it possible to observe the Galilean moons for long periods and that the higher-period orbits that bifurcate from the DROs offer additional insight into the motion of probes approaching/departing from the vicinities of the DROs. Further investigation of the invariant manifolds around higher-period orbits reveals the feasibility of utilizing the DRO family and the nearby unstable structures for multi-target exploration and low-energy transfer between the Galilean moons.

Application of delay embedding and recurrence analysis to a noisy nonlinear map for cyclic combustion dynamics of SI engines

A noisy nonlinear dynamical (NND) model available for cyclic variations in SI engines is examined using nonlinear time series methods. Heat release time series computed from this model is analyzed by employing bifurcation diagrams, return maps, recurrence plots, and recurrence quantitative analysis techniques. The analysis methodology is at first applied to a noisy Logistic map to optimize the necessary parameters, and later the same is used for the NND map. The presence of noise hides the local patterns of the dynamics and higher periodic orbits, making the bifurcation diagrams look fuzzy for both maps. An increase in the noise level and uncombusted residuals in the NND map advance the onset point of bifurcations toward a higher equivalence ratio. The application of the NND model is extended to represent the SI engine combustion dynamics for a higher amount of internal EGR as it is favored for cycle-resolved control due to its feedback period of one combustion cycle. The combined effect of lean charge and internal EGR leads to a more complex dynamical nature. The addition of a higher amount of internal EGR adds some complex chaotic-like regions in the bifurcation diagrams. For a low level of fluctuations in the NND map, more than four noisy periodic orbits are observed for such operating conditions. The geometry of phase space trajectories for the low-level noisy model data with a high level of internal EGR resembles a chaotic system. Bifurcation diagrams, recurrence plots, and quantitative recurrence analysis indicate an intermittent route to chaos at a highly lean and diluted charge. The presence of deterministic chaotic characteristics in combustion dynamics, along with effort-saving techniques of data acquisition and estimating combustion phasing, depicts the possibilities of practical application in active onboard control of engine systems.

Near rectilinear halo orbits and nearby higher-period dynamical structures: orbital stability and resonance properties

Near rectilinear halo orbits (NRHOs), a subset of the $$L_1$$ and $$L_2$$ halo orbit families, are strong candidates for a future inhabited facility in line with the goals for NASA’s long-term cislunar transportation network. Higher-period dynamical structures that bifurcate from the NRHO region of the $$L_2$$ halo family offer additional insight into motion departing from or arriving into the vicinity. Characteristics of the NRHOs and the higher-period orbits, including stability and eclipse-avoidance properties, are presented and their persistence is verified in a higher-fidelity ephemeris model.

Enhanced Energy Harvesting from Nonlinear Oscillators via Chaos Control

Abstract Modern electronic devices require less energy on-board and could be powered by energy harvested from the environment. Broadband vibration energy harvesting is one of the widely explored options for powering such devices. One such harvester that exploits this technique is the nonlinear inverted elastic pendulum. This system has a double-well potential and can undergo large amplitude inter-well oscillations. Piezoelectric patches, pasted near the base of the beam, convert these mechanical oscillations into electrical energy. However, the large amplitude responses can deteriorate over time into low energy chaotic oscillations, thereby reducing the harvested power. Under such conditions, forcing the system to follow a high energy orbit would improve the energy harvested from the system. However, it is not wise to invest a large control force to stabilize the high energy orbits. This manuscript aims to exploit the chaotic nature of the system to stabilize the unstable periodic orbits. The chaotic behavior of the system allows even small perturbations to alter its response dramatically. Taking advantage of this property, a low power controller, based on the method of Ott, Grebogi, and Yorke (OGY), is used to stabilize the unstable high energy periodic orbits. The control strategy is implemented on the linearized system, about the operating point corresponding to the chosen orbit. LQR is used to determine the optimum control force. The devised strategy is numerically simulated, which favors the implementation of OGY control to improve their energy harvesting capabilities.

A Sinusoidally Driven Lorenz System and Circuit Implementation

Another approach is developed for generating two-wing hyperchaotic attractor, four-wing chaotic attractor, and high periodic orbits such as period-14 from a sinusoidally driven based canonical Lorenz system. A sinusoidal function controller is introduced into a 3D autonomous Lorenz system, so that the abovementioned various hyperchaotic attractors, chaotic attractors, and high periodic orbits can be obtained, respectively, by adjusting the frequency of the sine function. In addition, an analog circuit and a digital circuit are also designed and implemented, with experimental results demonstrated. Both numerical simulations and circuit implementation together show the effectiveness of the proposed systematic methodology.

Analytic programming in the task of evolutionary synthesis of a controller for high order oscillations stabilization of discrete chaotic systems

This paper deals with the utilization of a symbolic regression tool, which is Analytic Programming (AP), together with two evolutionary algorithms, the Self-Organizing Migrating Algorithm (SOMA) and Differential Evolution (DE), for the synthesis of a new control law. This synthesized chaotic controller secures the stabilization of higher periodic orbits, which represent oscillations between several values of three selected discrete chaotic systems. Selected examples were: an artificially evolutionary synthesized system, logistic equation and Hénon map. The paper consists of the description of analytic programming as well as chaotic systems used, evolutionary techniques and the cost function.

Analysis and Modeling of an FFHC-Controlled DC–DC Buck Converter Suitable for Wide Range of Operating Conditions

To achieve the best optimized performance in terms of stability and dynamic behavior of power electronic converters, it is necessary to use a more advanced control technique and accurate mathematical model. This paper proposes a fixed-frequency hysteretic current (FFHC) controller that uses both sliding-mode control (SMC) technique and fixed-frequency current controller with a hysteresis band to achieve all properties of the variable structure controller. However, realizing such fixed-frequency sliding-mode controller using small-signal-averaged (SSA) model of the power converters and Utkin's equivalent control technique may not be valid for all conditions. We show that it can be applicable only when the fast-scale dynamics of the converter system is stable, which can be achieved successfully by analyzing the stability of the FFHC-controlled buck converter using Filippov method and Floquet theory. The regions of stability are then presented to show the domains of existence of nominal period-1 and higher periodic orbits in 2-D parameter space. We also demonstrate how to derive the equivalent control law from the modified tristate converter topology to design the controller. Finally, the experimental results are presented to validate the effectiveness of this hybrid FFHC controller.

Intermittency and bifurcation in SEPICs under voltage-mode control

A stroboscopic map for voltage-controlled single ended primary inductor converter (SEPIC) with pulse width modulation (PWM) is presented, where low-frequency oscillating phenomena such as quasi-periodic and intermittent quasi-periodic bifurcations occurring in the system are captured by numerical and experimental methods. According to bifurcation diagrams and nonlinear dynamical theory, the characteristics of the low-frequency oscillation and the mechanism for the appearance of the low-frequency oscillation are investigated. It is shown that as the controller parameter varies, the change in the conduction mode takes place from the continuous conduction mode (CCM) under the originally stable period one and high periodic orbits to the intermittent changes between CCM and discontinuous conduction mode (DCM), which may be related to the losing stability of the system and brought the system to exhibiting low-frequency oscillating behaviour in the time domain. Moreover, the occurrence of the intermittent quasi-periodic oscillation reflects that the system undergoes a Neimark–Sacker bifurcation.

The Dynamics of Regularized Discontinuous Maps with Applications to Impacting Systems

One-dimensional piecewise-smooth discontinuous maps (maps with gaps) are known to have surprisingly rich dynamics, including periodic orbits with very high period and bifurcation diagrams showing period-adding or period-incrementing behavior. In this paper we study a new class of maps, which we refer to as regularized one-dimensional discontinuous maps, because they give very similar dynamics to discontinuous maps and closely approximate them, yet are continuous. We show that regularized discontinuous maps arise naturally as limits of higher dimensional continuous maps when we study the global dynamics of two nonsmooth mechanical applications: the cam-follower system and the impact oscillator. We review the dynamics of discontinuous maps, study the dynamics of regularized discontinuous maps, and compare this to the behavior of the two mechanical applications which we model. We show that we observe period-adding and period-incrementing behavior in these two systems respectively but that the effect of the regularization is to lead to a progressive loss of the higher period orbits.

An open-plus-closed-loop control for chaotic Mathieu-Duffing oscillator

By using the idea of open-plus-closed-loop(OPCL) control, a controller composed of an external excitation and a linear feedback is designed to entrain chaotic trajectories of Mathieu-Duffing oscillator to its periodic and higher periodic orbits. The global basin of entrainment of this open-plus-closed-loop control is proved by combining the Lyapunov stability theory with a comparative theorem of initial value problems for second-order ordinary differential equations. Numerical simulations are performed to verify the theoretical results.

Ordered and chaotic Bohmian trajectories

We discuss the issue of ordered and chaotic trajectories in the Bohmian approach of Quantum Mechanics from points of view relevant to the methods of Celestial Mechanics. The Bohmian approach gives the same results as the orthodox (Copenhagen) approach, but it considers also underlying trajectories guided by the wave. The Bohmian trajectories are rather different from the corresponding classical trajectories. We give examples of a classical chaotic system that is ordered quantum-mechanically and of a classically ordered system that is mostly chaotic quantum mechanically. Then we consider quantum periodic orbits and ordered orbits, that can be represented by formal series of the “third integral” type, and we study their asymptotic properties leading to estimates of exponential stability. Such orbits do not approach the “nodal points” where the wavefunction ψ vanishes. On the other hand, when an orbit comes close to a nodal point, chaos is generated in the neighborhood of a hyperbolic point (called X-point). The generation of chaos is maximum when the X-point is close to the nodal point. Finally we remark that high order periodic orbits may behave as “effectively ordered” or “effectively chaotic” for long times before reaching the period.

Dynamics of a nearly symmetrical piecewise linear oscillator close to grazing incidence: Modelling and experimental verification

Experimental studies and mathematical modelling have been carried out for a nearly symmetrical piecewise linear oscillator to examine the bifurcation scenarios close to grazing. Higher period responses are found after grazing, although the period adding windows predicted as a generic feature of one-sided impacting systems are not observed. It appears that the presence of the second high stiffness spring stabilises additional periodic orbits. The global solution for a piecewise smooth model is developed by stitching locally valid maps. For the symmetrical case the highest period of response is three, if asymmetry in the gap and/or stiffness is introduced then higher periodic orbits are observed. Only small asymmetries are required to achieve a good correspondence with experiments. Further examination shows that many attractors are not stable to even small changes in the symmetry of the system.

Dynamics of a piecewise linear map with a gap

In this paper, we consider periodic solutions of discontinuous non-smooth maps. We show how the fixed points of a general piecewise linear map with a discontinuity (‘a map with a gap’) behave under parameter variation. We show in detail all the possible behaviours of period 1 and period 2 solutions. For positive gaps, we find that period 2 solutions can exist independently of period 1 solutions. Conversely, for negative gaps, period 1 and period 2 solutions can coexist. Higher periodic orbits can also exist and be stable and we give several examples of how these solutions behave under parameter variation. Finally, we compare our results with those of Jain & Banerjee (Jain & Banerjee 2003 Int. J. Bifurcat. Chaos 13 , 3341–3351) and Banerjee et al . (Banerjee et al . 2004 IEEE Trans. Circ. Syst. II 51 , 649–654) and explain their numerical simulations.

CONTROLLING CHAOS IN A CHAOTIC NEURON MODEL

In this paper we investigate several methods for controlling chaos in Aihara's chaotic neuron model. We first discuss the stability of exponential feedback control method for this model. To obviate predetermining the unstable periodic orbits of the system, two other methods are developed. We analyze why the conventional delayed feedback control method cannot be employed here, and then give a modified form for recursive delayed feedback control and apply it to control chaos in this model. To obtain high-periodic orbits more easily, a delayed exponential feedback control method is proposed, by which we can obtain different periodic orbits by changing parameters. Computer simulations show good control effects and robustness against noise.

A CHARACTERISTIC OF THE TRANSITION TO CHAOS VIA TYPE V INTERMITTENCY

In the transition to chaos via type V intermittency, when a periodic motion loses stability, there may be no intermediate higher-period orbits displayed before chaos or a sequential such orbits that form a staircase. The staircase may take a form of a conventional devil's staircase, a multiple devil's staircase or a seemingly irregular form. This article briefly presents typical examples for these situations.

Analysis and Interpretation of the Delta 180 Collision Experiment in Space

In a landmark experiment in 1986, the payload of a Delta 180 launch vehicle was made to collide with its second stage rocket. The collision produced two separate debris clouds without any significant center-of-mass cloud. The inclination and plane change angle distributions of the fragments in the two clouds exhibited Gaussian patterns, with the payload cloud having a considerably greater dispersion than the rocket cloud. Several fragments strewn between the two clouds showed some momentum transfer and hence originated from the contact area. The centers of the two clouds moved away from each other in apparent compliance with momentum conservation. The three orthogonal components of the velocity perturbations received by the fragments are calculated and the angular distributions of the fragments obtained. The total velocity perturbations could be fitted with a Maxwellian distribution whereas its components generally followed Gaussian trends. Fragments from the contact areas, including those thrown into high period orbits, generally fell outside these envelopes. A vast majority of the fragments from both clouds received velocity perturbations below the horizontal plane, thus indicating a loss of translational kinetic energy due to the fragmentation. Three preferred directions in the fragment spread in the horizontal plane suggest that the payload hit the rocket body at an angle from behind.

Local and global control of high-period unstable orbits in reversible maps.

We study the nonlinear dynamics of a complex system, described by a two-dimensional reversible map. The phase space of this map exhibits elements typical of Hamiltonian systems (stability islands) as well as of dissipative systems (attractor). Due to the interaction between the stability islands and the attractor, the transition to chaos in this system occurs through the collapse of the stability island and stochastization of the limiting-cycles orbits. We show how to apply the method of discrete parametric control to stabilize unstable high-period orbits. To achieve highly efficient control we introduce the concepts of local and global control. These concepts are useful in situations where there are "dangerous" points on the target orbit, i.e., the points where the probability of breakdown of control is high. As a result, the dangerous points turn out to be much more sensitive to external noise than other points on the orbit, and only the dangerous points determine how effective the control is.

From template analysis to generating partitions: I: Periodic orbits, knots and symbolic encodings

We present a detailed algorithm to construct symbolic encodings for chaotic attractors of three-dimensional flows. It is based on a topological analysis of unstable periodic orbits embedded in the attractor and follows the approach proposed by Lefranc et al. [Phys. Rev. Lett. 73 (1994) 1364]. For each orbit, the symbolic names that are consistent with its knot-theoretic invariants and with the topological structure of the attractor are first obtained using template analysis. This information and the locations of the periodic orbits in the section plane are then used to construct a generating partition by means of triangulations. We provide numerical evidence of the validity of this method by applying it successfully to sets of more than 1500 periodic orbits extracted from numerical simulations, and obtain partitions whose border is localized with a precision of 0.01%. A distinctive advantage of this approach is that the solution is progressively refined using higher-period orbits, which makes it robust to noise, and suitable for analyzing experimental time series. Furthermore, the resulting encodings are by construction consistent in the corresponding limits with those rigorously known for both one-dimensional and hyperbolic maps.

Global analysis of periodic orbit bifurcations in coupled Morse oscillator systems: time-reversal symmetry, permutational representations and codimension-2 collisions.

In this paper we study periodic orbit bifurcation sequences in a system of two coupled Morse oscillators. Time-reversal symmetry is exploited to determine periodic orbits by iteration of symmetry lines. The permutational representation of Tsuchiya and Jaffe is employed to analyze periodic orbit configurations on the symmetry lines. Local pruning rules are formulated, and a global analysis of possible bifurcation sequences of symmetric periodic orbits is made. Analysis of periodic orbit bifurcations on symmetry lines determines bifurcation sequences, together with periodic orbit periodicities and stabilities. The correlation between certain bifurcations is explained. The passage from an integrable limit to nointegrability is marked by the appearance of tangent bifurcations; our global analysis reveals the origin of these ubiquitous tangencies. For period-1 orbits, tangencies appear by a simple disconnection mechanism. For higher period orbits, a different mechanism involving 2-parameter collisions of bifurcations is found. (c) 1999 American Institute of Physics.