Unstable Periodic Orbits Research Articles

Hidden and unstable periodic orbits as a result of homoclinic bifurcations in the Suarez–Schopf delayed oscillator and the irregularity of ENSO

We revisit the classical Suarez–Schopf delayed oscillator. Special attention is paid to the region of linear stability in the space of parameters. By means of the theory of inertial manifolds developed in our adjacent papers, we provide analytical–numerical evidence for the existence of two-dimensional inertial manifolds in the model. This allows to suggest a complete qualitative description of the dynamics in the region of linear stability. We show that there are two subregions corresponding to the existence of hidden or self-excited attracting periodic orbits. These subregions must be separated by a curve on which homoclinic “figure eights”, bifurcating into a single one or a pair of unstable periodic orbits, should exist. We relate the observed hidden oscillations and homoclinics to the irregularity theories of ENSO and provide numerical evidence that chaotic behavior may appear if a small periodic forcing is applied to the model. We also use parameters from the Suarez–Schopf model to discover hidden and self-excited asynchronous periodic regimes in a ring array of coupled lossless transmission lines studied by J. Wu and H. Xia.

Intermittent chaotic spiking in the van der Pol–FitzHugh–Nagumo system with inertia

The van der Pol–FitzHugh–Nagumo neuron model with inertia was shown to exhibit a chaotic mixed-mode dynamics composed of large-amplitude spikes separated by an irregular number of small-amplitude chaotic oscillations. In contrast to the standard 2D van der Pol–FitzHugh–Nagumo model driven by noise, the inter-spike intervals distribution displays a complex arrangement of sharp peaks related to the unstable periodic orbits of the chaotic attractor. For many ranges of parameters controlling the excitability of the system, we observe that chaotic mixed-mode states consist of lapses of nearly regular spiking interleaved by others of highly irregular one. We explore here the emergence of these structures and show their correspondence to the intermittent transitions to chaos. In fact, the average residence times in the nearly-periodic firing state, obey the same scaling law – as a function of the control parameter – than the one at the onset of type I intermittency for dynamical systems in the vicinity of a saddle node bifurcation. We hypothesize that this scenario is also present in a variety of slow-fast neuron models characterized by the coexistence of a two-dimensional fast manifold and a one-dimensional slow one.

Safe Lane-Keeping With Feedback Delay: Bifurcation Analysis and Experiments

A lane-keeping controller for automobiles is analyzed in this letter, with the consideration of time delay in the feedback loop. Unstable periodic orbits are identified inside the linearly stable domain of control gains. Based on the amplitude of these unstable solutions, safe parameter zones are identified, where the closed loop system is robust against perturbations, i.e., where the basin of attraction of the stable equilibrium is sufficiently large. Sensitivity to perturbations in different regions of linearly stable control gains is demonstrated by a series of real-vehicle experiments. Finally, modifications of the control law are proposed that lead to significant improvements in terms of the robustness against perturbation for the controlled vehicle, achieving global stability in the practical sense.

Detecting unstable periodic points of chaotic maps by stability transformation of reservoir

Detecting unstable periodic orbits in chaotic systems based on the time series is a fundamental problem in nonlinear dynamics, but it often becomes extremely challenging one. In this study, we propose a new approach for detecting unstable periodic points using reservoir computing and stability transformation method. We connects reservoir computing, which is a well-known machine learning technique, and stability transformation method, which can detect unstable periodic points in chaotic dynamical systems, to perform unstable periodic points detection in a data-driven and model-free process. In this paper, we use an example of the Hénon map to demonstrate detecting unstable fixed point and unstable 2-periodic points.

DYNAMIC ANALYSIS OF A DISCRETE AMENSALISM MODEL WITH ALLEE EFFECT

This paper concerns with a discretization of a continuous-time amensalism model with Allee effect on the first species. Compared with the continuous analog, the discrete system has different and much richer dynamical behaviors. First, we obtain the existence of fixed points and their local stability. Then we confirm the occurrence of fold bifurcation and period doubling bifurcation by using the center manifold theorem and bifurcation theory. Followed is a hybrid control strategy to control the period-doubling bifurcation and stabilize unstable periodic orbits embedded in the complex attractor. Numerical simulations indicate that Allee effect is beneficial to the stability of the first species to a certain extent. Moreover, when the first species is affected by Allee effect, solutions can approach the corresponding fixed point more quickly.

Rigorous numerical study of the Colpitts oscillator with an exponential nonlinearity.

The dynamics of the Colpitts oscillator with an exponential nonlinearity is investigated using rigorous interval arithmetic based tools. The existence of various types of periodic attractors is proved using the interval Newton method. The main results involve the chaotic case for which a trapping region for the associated return map is constructed and a rigorous lower bound for the value of the topological entropy is computed, thus proving that the system is chaotic in the topological sense. A systematic search for unstable periodic orbits embedded in the chaotic attractor is carried out and the results are used to obtain an accurate approximation of the topological entropy of the system.

Nonlinear stochastic dynamics research on a Lorenz system with white Gaussian noise based on a quasi‐potential approach

AbstractEnvironmental noise can lead to complex stochastic dynamical behaviors in nonlinear systems. In this paper, a Lorenz system with the parameter region with two stable fixed points and a chaotic saddle subject to white Gaussian noise is investigated as an example. Noise‐induced phenomena, such as noise‐induced quasi‐cycle, three‐state intermittency, and chaos, are observed. In the intermittency process, the optimal path used to describe the transition mechanism is calculated and confirmed to pass through an unstable periodic orbit, a chaotic saddle, a saddle point, and a heteroclinic trajectory in an orderly sequence using generalized cell mapping with a digraph method constructively. The corresponding optimal fluctuation forces are delineated to uncover the effects of noise during the transition process. Then the process will switch frequently between the attractors and the chaotic saddle as noise intensity increased further, that is, noise induced chaos emerging. A threshold noise intensity is defined by stochastic sensitivity analysis when a confidence ellipsoid is tangent to the stable manifold of the periodic orbit, which agrees with the simulation results. It is finally reported that these results and methods can be generalized to analyze the stochastic dynamics of other nonlinear mechanical systems with similar structures.

Model-free continuation of periodic orbits in certain nonlinear systems using continuous-time adaptive control

This paper generalizes recent results by the authors on noninvasive model-reference adaptive control designs for control-based continuation of periodic orbits in periodically excited linear systems with matched uncertainties to a larger class of periodically excited nonlinear systems with matched uncertainties and known structure. A candidate adaptive feedback design is also proposed in the case of scalar problems with unmodeled nonlinearities. In the former case, rigorous analysis shows guaranteed performance bounds for the associated prediction and estimation errors. Together with an assumption of persistent excitation, there follows asymptotic convergence to periodic responses determined uniquely by an a priori unknown periodic reference input and independent of initial conditions, as required by the control-based continuation paradigm. In particular, when the reference input equals the sought periodic response, the steady-state control input vanishes. Identical conclusions follow for the case of scalar dynamics with unmodeled nonlinearities, albeit with slow rates of convergence. Numerical simulations validate the theoretical predictions for individual parameter values. Integration with the software package coco demonstrates successful continuation along families of stable and unstable periodic orbits with a minimum of parameter tuning. The results expand the envelope of known noninvasive feedback strategies for use in experimental model validation and engineering design.

A New Variable-Boostable 3D Chaotic System with Hidden and Coexisting Attractors: Dynamical Analysis, Periodic Orbit Coding, Circuit Simulation, and Synchronization

The study of hidden attractors plays a very important role in the engineering applications of nonlinear dynamical systems. In this paper, a new three-dimensional (3D) chaotic system is proposed in which hidden attractors and self-excited attractors appear as the parameters change. Meanwhile, asymmetric coexisting attractors are also found as a result of the system symmetry. The complex dynamical behaviors of the proposed system were investigated using various tools, including time-series diagrams, Poincaré first return maps, bifurcation diagrams, and basins of attraction. Moreover, the unstable periodic orbits within a topological length of 3 in the hidden chaotic attractor were calculated systematically by the variational method, which required six letters to establish suitable symbolic dynamics. Furthermore, the practicality of the hidden attractor chaotic system was verified by circuit simulations. Finally, offset boosting control and adaptive synchronization were used to investigate the utility of the proposed chaotic system in engineering applications.

Unstable periodic orbits analysis in the Qi system

We use the variational method to extract the short periodic orbits of the Qi system within a certain topological length. The chaotic dynamical behaviors of the Qi system with five equilibria are analyzed by the means of phase portraits, Lyapunov exponents, and Poincaré maps. Based on several periodic orbits with different sizes and shapes, they are encoded systematically with two letters or four letters for two different sets of parameters. The periodic orbits outside the attractor with complex topology are discovered by accident. In addition, the bifurcations of cycles and the bifurcations of equilibria in the Qi system are explored by different methods respectively. In this process, the rule of orbital period changing with parameters is also investigated. The calculation and classification method of periodic orbits in this study can be widely used in other similar low-dimensional dissipative systems.

The unpredictable nature of bubble evolution

Unpredictable dynamics arising from a sensitivity to initial conditions is commonly associated with chaos. We demonstrate how similar unpredictability manifests in a nonlinear system that possesses a large number of long-term outcomes, namely the propagation of an air bubble within a viscous fluid-filled channel. The system under investigation supports various stable states of single-bubble propagation. In addition, bubbles can readily break up during their propagation. Upon subjecting steadily-propagating bubbles to finite-amplitude perturbations in the form of localised channel constrictions, we identify localised regions of the driving flow rate for which the resulting evolutions are unpredictable. Visibly-indistinguishable bubbles are observed to evolve towards a multitude of long-term outcomes, including each of the stable states available to the initial bubble and various states of permanently-changed bubble topology. By combining high-precision experimental results with simulations of a depth-averaged lubrication model of the system, we determine that this behaviour is driven by a sensitive dependence on initial conditions within the vicinity of an unstable periodic orbit.

Invariant tori in dissipative hyperchaos.

One approach to understand the chaotic dynamics of nonlinear dissipative systems is the study of non-chaotic yet dynamically unstable invariant solutions embedded in the system's chaotic attractor. The significance of zero-dimensional unstable fixed points and one-dimensional unstable periodic orbits capturing time-periodic dynamics is widely accepted for high-dimensional chaotic systems, including fluid turbulence, while higher-dimensional invariant tori representing quasiperiodic dynamics have rarely been considered. We demonstrate that unstable 2-tori are generically embedded in the hyperchaotic attractor of a dissipative system of ordinary differential equations; tori can be numerically identified via bifurcations of unstable periodic orbits and their parameteric continuation and characterization of stability properties are feasible. As higher-dimensional tori are expected to be structurally unstable, 2-tori together with periodic orbits and equilibria form a complete set of relevant invariant solutions on which to base a dynamical description of chaos.

Cupolets in a chaotic neuron model.

This paper reports the first finding of cupolets in a chaotic Hindmarsh-Rose neural model. Cupolets (chaotic, unstable, periodic, orbit-lets) are unstable periodic orbits that have been stabilized through a particular control scheme by applying a binary control sequence. We demonstrate different neural dynamics (periodic or chaotic) of the Hindmarsh-Rose model through a bifurcation diagram where the external input current, I, is the bifurcation parameter. We select a region in the chaotic parameter space and provide the results of numerical simulations. In this chosen parameter space, a control scheme is applied when the trajectory intersects with either of the two control planes. The type of the control is determined by a bit in a binary control sequence. The control is either a small microcontrol (0) or a large macrocontrol (1) that adjusts the future dynamics of the trajectory by a perturbation determined by the coding function r ( x ). We report the discovery of many cupolets with corresponding control sequences and comment on the differences with previously reported cupolets in the double scroll system. We provide some examples of the generated cupolets and conclude by discussing potential implications for biological neurons.

Qualitative changes in bifurcation structure for soft vs hard impact models of a vibro-impact energy harvester.

Vibro-impact phenomena in engineering systems, considered an adverse effect in some settings, are an intrinsic part of the mechanism in others. In energy harvesting, a vibro-impact component is often intentionally introduced to increase the power output or the system's bandwidth. The impacts can be treated as "hard" for instantaneous impacts or "soft" for compliant materials. Since both types of models exhibit complex dynamics, a comparison is non-trivial. We develop a soft impact model for a vibro-impact energy harvester, calibrating it with the relevant hard impact model for large stiffness, and systematically compare the different phenomena and dynamics in various compliant regimes. Numerical results are used in two different parametric analyses, considering the bifurcation diagrams in terms of device size and external forcing parameters. Varying the natural frequency of the membranes that form the impact boundaries, we observe shifts in the bifurcation structure that promote period-1 orbits for increased softness parameters, often generating higher power output, but also introducing parameter sensitivities for increased softness. Complementary analytical results reveal unstable periodic orbits and co-existing behaviors, potentially missed by computational methods, that can influence the bifurcation structure and in turn the energy output. A non-dimensional formulation highlights the significance of ratios of external and natural frequencies in delineating soft and hard impact scenarios parametrically. The soft impact model exhibits new symmetry breaking bifurcations related to key quantities that characterize the soft impact dynamics, such as the effective restitution coefficients, the impact phase, and the contact time interval, not captured by hard impact models.

The Influence of Asymmetry on the Dynamics Associated with a Caldera Potential Energy Surface

In this paper, we analyze the influence of asymmetry on a Caldera potential energy surface. We first study the effect of asymmetry on the structure of the periodic orbit dividing surfaces associated with the unstable periodic orbits of the higher energy index-1 saddles. Then we detect a new type of dynamical matching due to the influence of the asymmetry in the Caldera potential energy surface. Finally, we present the phase space mechanism for this new type of dynamical matching using Lagrangian Descriptors.

Search for invariant sets of the generalized tent map

This paper describes a predictive control method to search for unstable periodic orbits of the generalized tent map. The invariant set containing periodic orbits is a repelling set with a complicated Cantor-like structure. Therefore, a simple local stabilization of the orbit may not be enough to find a periodic orbit, due to the small measure of the basin of attraction. It is shown that for certain values of the control parameter, both the local behaviour and the global behaviour of solutions change in the controlled system; in particular, the invariant set enlarges to become an interval or the entire real axis. The computational particularities of using the control system are considered, and necessary conditions for the orbit to be periodic are given. The question of local asymptotic stability of subcycles of the controlled system's stable cycles is fully investigated, and some statistical properties of the subset of the classical Cantor middle thirds set that is determined by the periodic points of the generalized tent map are described.

Stabilizing unstable periodic orbits in large stability domains with dynamic time-delayed feedback control

Dynamic time-delayed feedback control (DDFC) is applied to stabilize the unstable periodic orbits (UPOs) of chaotic systems in large stability domains. The stability domain is defined as certain areas of the parameter space of the feedback strength, in which the UPOs are stabilized. The control effect of the DDFC with a second-order controller system is investigated by considering two control objectives: to broaden the stability domain of the controlled UPOs and to minimize the modulus of the largest Floquet multiplier, which leads to a multi-objective optimization problem (MOP). The MOP is solved with the genetic algorithm. Case studies indicate that the control effect of the DDFC is significantly better than that of original time-delayed feedback control. The DDFC can stabilize the UPOs in a large stability domain and with a small modulus of the largest Floquet multiplier when the adjustable parameters of the controller system are properly designed.

The phase space structure in the vicinity of vertical Lyapunov orbits around L1,2 in a barred galaxy model

ABSTRACT We study the phase space structure and the orbital diffusion from the vicinity of the vertical Lyapunov periodic orbits around the unstable Lagrangian points L1,2 in a 3D barred galaxy model. By perturbing the initial conditions of these periodic orbits, we detected the following five types of orbital structures in the 4D spaces of section: (i) Ring-like structures, sticky for large time intervals to the unstable invariant manifolds of the simple and double unstable vertical Lyapunov periodic orbits. (ii) 2D tori belonging to quasi-periodic orbits around stable periodic orbits existing in the region. They are associated either with vertical stable periodic orbits around L4,5 or with ‘stable anomalous’ periodic orbits. (iii) Orbits sticky for large time intervals to these tori, forming ‘sticky tori’, before they slowly depart from them. (iv) Clouds of points that have a strong chaotic behaviour. Such clouds of consequents have slow diffusion speeds, because they are hindered by the presence of the tori around the ‘stable anomalous’ periodic orbits. (v) Toroidal zones consisting of points that stick for long time on the unstable invariant manifolds of the ‘unstable anomalous’ periodic orbits. By continuing the integration, we find that eventually they become strongly chaotic, retaining however small diffusion speeds, due to the presence of the tori around the stable anomalous periodic orbits.

Nonlinear analysis of the delayed tyre model with control-based continuation

In this study, the numerical bifurcation analysis of a shimmying wheel is performed with a non-smooth, time-delayed model of the tyre-ground contact. This model is capable of reproducing the bistable behaviour often observed in experiments: a stable equilibrium and a stable periodic orbit coexisting for the same set of system parameters, that the simpler quasi-steady tyre models fail to capture. In the bistable parameter domain, there also exists an unstable periodic orbit within the separatrix between the domains of attractions of the two stable steady-state solutions. Although this solution never appears in a real-life system, one may still gain valuable information from tracing it as it gives an indication about the level of perturbation that would drive the system from one stable solution to the other. However, the complexity of the laws governing partial sticking and sliding in the tyre-ground contact makes the numerical bifurcation analysis with the traditional, collocation-based techniques infeasible. Instead, this study is based on numerical simulations and the technique of control-based continuation (CBC) to track the stable and unstable periodic solutions of the system allowing for the assessment of the accuracy of the non-smooth, delayed tyre model in replicating the dynamics observed in experiments. In the meantime, the physics-based model provides an insight into the relationship between the sticking and sliding regions appearing in the tyre-ground contact and the global dynamics of the system.

Relativistic quantum scarring, spin-induced phase, and quantization in a symmetric Dirac billiard system

Thirty-five years ago, Sir Michael Berry and his collaborator Mondragon studied the behaviors of neutrino, a massless relativistic quantum particle, in a classically chaotic billiard—the neutrino billiard problem. To celebrate Sir Michael Berry’s eightieth birthday, here we report results on the role of geometric symmetries of the billiard system in relativistic quantum scarring. In particular, we investigate a Dirac billiard system with a four-fold rotational symmetry whose classical dynamics are fully chaotic. The system is described by the massless Dirac equation in the fundamental domain that consists of one fourth of the full billiard, with proper boundary conditions on the symmetry lines to preserve the physical properties under the symmetry operations. We show that the relativistic quantum characteristics of spin induced phase play a fundamental role in the quantum behaviors of the Dirac particle in the billiard. We find that the peaks in the length spectra are due to the interference of states circling the fundamental domain orbits (FDOs) in opposite propagating directions, which can be constructive or destructive depending on the accumulated phases. In addition, we derive the quantization conditions of the scarring states about the unstable periodic orbits within the fundamental domain from the phase along the FDOs. Our work is a vivid demonstration that relativistic quantum scarring, or more generally quantum manifestations of classical chaos, can be fully understood by analyzing the behaviors of the geometric phase—a powerful approach in modern physics pioneered by Sir Michael Berry.