Smale Horseshoe Research Articles

Single-pulse Chaotic Dynamics in Suspended Cables with Parametric and External Excitations

The chief aim of the present work is to investigate the single-pulse chaotic dynamics for suspended cables with principal parametric and primary resonant excitation. Using higherdimensional Melnikov theory, explicit conditions for the existences of single-pulse Silnikov-type homoclinic orbits are obtained, which imply that chaotic motions in the sense of Smale horseshoes may occur for this system. Finally, numerical results are presented to verify these analytical predictions.

Analysis on global and chaotic dynamics of nonlinear wave equations for truss core sandwich plate

This paper presents the study on the chaotic wave and chaotic dynamics of the nonlinear wave equations for a simply supported truss core sandwich plate combined with the transverse and in-plane excitations. Based on the governing equation of motion for the simply supported sandwich plate with truss core, the reductive perturbation method is used to simplify the partial differential equation. According to the exact solution of the unperturbed equation, two different kinds of the topological structures are derived, which one structure is the resonant torus and another structure is the heteroclinic orbit. The characteristic of the singular points in the neighborhood of the resonant torus for the nonlinear wave equation is investigated. It is found that there exists the homoclinic orbit on the unperturbed slow manifold. The saddle-focus type of the singular point appears when the homoclinic orbit is broken under the perturbation. Additionally, the saddle-focus type of the singular point occurs when the resonant torus on the fast manifold is broken under the perturbation. It is known that the dynamic characteristics are well consistent on the fast and slow manifolds under the condition of the perturbation. The Melnikov method, which is called the first measure, is applied to study the persistence of the heteroclinic orbit in the perturbed equation. The geometric analysis, which is named the second measure, is used to guarantee that the heteroclinic orbit on the fast manifold comes back to the stable manifold of the saddle on the slow manifold under the perturbation. The theoretical analysis suggests that there is the chaos for the Smale horseshoe sense in the truss core sandwich plate. Numerical simulations are performed to further verify the existence of the chaotic wave and chaotic motions in the nonlinear wave equation. The damping coefficient is considered as the controlling parameter to study the effect on the propagation property of the nonlinear wave in the sandwich plate with truss core. The numerical results confirm the validity of the theoretical study.

Oscillatory motions in restricted N-body problems

We consider the planar restricted N-body problem where the N−1 primaries are assumed to be in a central configuration whereas the infinitesimal particle escapes to infinity in a parabolic orbit. We prove the existence of transversal intersections between the stable and unstable manifolds of the parabolic orbits at infinity which guarantee the existence of a Smale's horseshoe. This implies the occurrence of chaotic motions, namely the oscillatory motions, that is, orbits for which the massless particle leaves every bounded region but it returns infinitely often to some fixed bounded region. Our achievement is based in an adequate scaling of the variables which allows us to write the Hamiltonian function as the Hamiltonian of the Kepler problem plus higher-order terms that depend on the chosen configuration. We compute the Melnikov function related to the first non-null perturbative term and characterize the cases where it has simple zeroes. Concretely, for some combinations of the configuration parameters, i.e. mass values and positions of the primaries, and for a specific value of a parameter related to the angular momentum vector, the Melnikov function vanishes, otherwise it has simple zeroes and the transversality condition is satisfied. When the Melnikov function corresponding to the principal part of the perturbation is zero we compute the next non-zero Melnikov function proving that it has simple zeroes. The theory is illustrated for various cases of restricted N-body problems, including the circular restricted three-body problem. No restrictions on the mass parameters are assumed.

Control of an electromechanical pendulum subjected to impulsive disturbances using the Melnikov theory approach

The dynamics of an electromechanical pendulum that collides with an external moving mass is considered. The Melnikov function is derived to determine the effects of periodic collisions on the threshold condition for the appearance of Smale horseshoes chaos in the system. In order to counterbalance the action of the collision, a pulse-like periodic controller is used and the results show the efficiency of the controller to reduce the distortions due to collision and change the parameters boundary delineating the chaotic domain.

Smale Horseshoes and Symbolic Dynamics in the Buck–Boost DC–DC Converter

DC–DC converter, which is used to change and control voltage magnitude, is a really important part of the power electronic conditioning systems. Study of nonlinear dynamics and chaos in power electronics are becoming increasingly significant in the power quality and stability of distributed power generation systems. In this paper, the mechanisms that give rise to chaotic dynamical in the buck–boost power converter is studied. The Poincare map of a buck–boost converter system possesses the invariant manifolds having a delightfully complicated structure. Based on the Smale–Birkhoff homoclinic theorem, the structure of invariant manifolds of a fixed point indicates the presence of Smale horseshoe. The horseshoe map of the system is topologically conjugate with the shift map of the symbolic dynamics. The system can be identified with the Smale horseshoe by means of its symbolic descriptions. Furthermore, since the Smale horseshoe map implies sensitive dependence on the initial condition, homoclinic intersection of stable and unstable manifolds may be applied to illustrate the existence of chaos.

Cascades of Global Bifurcations and Chaos near a Homoclinic Flip Bifurcation: A Case Study

We study a homoclinic flip bifurcation of case~\textbf{C}, where a homoclinic orbit to a saddle equilibrium with real eigenvalues changes from being orientable to nonorientable. This bifurcation is of codimension two, and it is the lowest codimension for a homoclinic bifurcation of a real saddle to generate chaotic behavior in the form of (suspended) Smale horseshoes and strange attractors. We present a detailed numerical case study of how global stable and unstable manifolds of the saddle equilibrium and of bifurcating periodic orbits interact close to such bifurcation. This is a step forward in understanding the generic cases of homoclinic flip bifurcations, which started with the study of the simpler cases \textbf{A} and \textbf{B}. In a three-dimensional vector field due to Sandstede, we compute relevant bifurcation curves in the two-parameter bifurcation diagram near the central codimension-two bifurcation in unprecedented detail. We present representative images of invariant manifolds, computed with a boundary value problem setup, both in phase space and as intersection sets with a suitable sphere. In this way, we are able to identify infinitely many cascades of homoclinic bifurcations that accumulate on specific codimension-one heteroclinic bifurcations between an equilibrium and various saddle periodic orbits. Our computations confirm what is known from theory but also show the existence of bifurcation phenomena that were not considered before. Specifically, we identify the boundaries of the Smale--horseshoe region in the parameter plane, one of which creates a strange attractor that resembles the R\"{o}ssler attractor. The computation of a winding number reveals a complicated overall bifurcation structure in the wider parameter plane that involves infinitely many further homoclinic flip bifurcations associated with so-called homoclinic bubbles.

ON THE ONE-WAY SPECIFICATION PROPERTY AND LARGE DEVIATIONS FOR SYSTEMS WITH NON-DENSE ERGODIC MEASURES

We introduce a weaker form of the specification property, called the ‘one-way specification property', which holds for a broad class of systems whose ergodic measures are not dense in the set of invariant measures, including Smale's horseshoe map, DA map, and (symbolic spaces of) (–β)-transformations for Yrrap β > 1. As an application, we show that these three examples satisfy a level-2 large deviation principle with the normalized Lebesgue measure.

Global dynamics of a flexible asymmetrical rotor

Global dynamics of a flexible asymmetrical rotor resting on vibrating supports is investigated. Hamilton’s principle is used to derive the partial differential governing equations of the rotor system. The equations are then transformed into a discretized nonlinear gyroscopic system via Galerkin’s method. The canonical transformation and normal form theory are applied to reduce the system to the near-integrable Hamiltonian standard forms considering zero-to-one internal resonance. The energy-phase method is employed to study the chaotic dynamics by identifying the existence of multi-pulse jumping orbits in the perturbed phase space. In both the Hamiltonian and the dissipative perturbation, the homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are demonstrated. In the case of damping dissipative perturbation, the existence of generalized Silnikov-type multi-pulse orbits which are homoclinic to fixed points on the slow manifold is examined and the parameter region for which the dynamical system may exhibit chaotic motions in the sense of Smale horseshoes is obtained analytically. The global results are finally interpreted in terms of the physical motion of axis orbit. The present study indicates that the existence of multi-pulse homoclinic orbits provides a mechanism for how energy may flow from the high-frequency mode to the low-frequency mode when the rotor system operates near the first-order critical speed.

Sigma map dynamics and bifurcations

Some interesting variants of walking droplet based discrete dynamical bifurcations arising from diffeomorphisms are analyzed in detail. A notable feature of these new bifurcations is that, like Smale horseshoes, they can be represented by simple geometric paradigms, which markedly simplify their analysis. The two-dimensional diffeomorphisms that produce these bifurcations are called sigma maps or double sigma maps for reasons that are made manifest in this investigation. Several examples are presented along with their dynamical simulations.

Analytical threshold for chaos in a Duffing oscillator with delayed feedbacks

In this paper, the bifurcation and chaotic behaviors of a Duffing oscillator with delayed displacement and velocity feedbacks under harmonic excitation are investigated. The analytically necessary condition for the chaos in the sense of Smale horseshoes is established based on Melnikov method. It could be found that the delayed displacement and velocity feedbacks affect the necessary condition respectively, so that the analysis on the effects of the two kinds of feedbacks is presented individually. The effects of displacement feedback coefficient, velocity feedback coefficient and their time delays on the analytically necessary condition are analytically discussed. Furthermore, some typical numerical results, including the bifurcation diagrams, Poincaré maps and the largest Lyapunov exponents of the delayed oscillator are also presented and compared. The satisfactory qualitative agreement between the analytical and numerical results verify the correctness of the analytically necessary condition.

Quantifying the Role of Folding in Nonautonomous Flows: The Unsteady Double-Gyre

We analyze chaos in the well-known nonautonomous Double-Gyre system. A key focus is on folding, which is possibly the less-studied aspect of the “stretching+folding=chaos” mantra of chaotic dynamics. Despite the Double-Gyre not having the classical homoclinic structure for the usage of the Smale–Birkhoff theorem to establish chaos, we use the concept of folding to prove the existence of an embedded horseshoe map. We also show how curvature of manifolds can be used to identify fold points in the Double-Gyre. This method is applicable to general nonautonomous flows in two dimensions, defined for either finite or infinite times.

Global bifurcations and chaotic motions of a flexible multi-beam structure

Global bifurcations and multi-pulse chaotic motions of flexible multi-beam structures derived from an L-shaped beam resting on a vibrating base are investigated considering one to two internal resonance and principal resonance. Base on the exact modal functions and the orthogonality conditions of global modes, the PDEs of the structure including both nonlinear coupling and nonlinear inertia are discretized into a set of coupled autoparametric ODEs by using Galerkin’s technique. The method of multiple scales is applied to yield a set of autonomous equations of the first order approximations to the response of the dynamical system. A generalized Melnikov method is used to study global dynamics for the “resonance case”. The present analysis indicates multi-pulse chaotic motions result from the existence of Šilnikov’s type of homoclinic orbits and the critical parameter surface under which the system may exhibit chaos in the sense of Smale horseshoes are obtained. The global results are finally interpreted in terms of the physical motion of such flexible multi-beam structure and the dynamical mechanism on chaotic pattern conversion between the localized mode and the coupled mode are revealed.

Dynamical modeling and multi-pulse chaotic dynamics of cantilevered pipe conveying pulsating fluid in parametric resonance

The dynamical modeling, multi-pulse orbits and chaotic dynamics of cantilevered pipe conveying pulsating fluid with harmonic external force are studied using the energy-phase method for the first time. The nonlinear geometric deformation of the pipe and the Kelvin constitutive relation of the pipe material are considered. The nonlinear governing equation of motion for cantilevered pipe conveying pulsating fluid is determined using Hamilton principle. The four-dimensional averaged equation in the case of primary parametric resonance-1/2 subharmonic resonance and 1:2 internal resonance is obtained by directly using the method of multiple scales and the Galerkin approach to the partial differential governing equation of motion for cantilevered pipe conveying pulsating fluid. From the averaged equation, the normal form theory is used to find the explicit formulas of normal form. Based on normal form obtained here, the energy-phase method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics of cantilevered pipe conveying pulsating fluid. The analysis of global dynamics indicates that the multi-pulse jumping orbits exist in the perturbed phase space of the averaged equation. Based on the averaged equation, the chaotic motions and the multi-pulse orbits for the cantilevered pipe are found by using numerical simulations. The results described above indicate the existence of chaos in the Smale horseshoe sense for cantilevered pipe conveying pulsating fluid. It has been concluded that it is dangerous to induce the chaotic vibrations of the pipes conveying pulsating fluid because the amplitudes of the chaotic vibrations are larger than those of the periodic motions.

Multi-pulse chaotic motions of functionally graded truncated conical shell under complex loads

The global bifurcations and multi-pulse orbits of an aero-thermo-elastic functionally graded material (FGM) truncated conical shell under complex loads are investigated with the case of 1:2 internal resonance and primary parametric resonance. The method of multiple scales is utilized to obtain the averaged equations. Based on the averaged equations obtained, the normal form theory is employed to find the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues. The energy-phase method developed by Haller and Wiggins is used to analyze the multi-pulse homoclinic bifurcations and chaotic dynamics of the FGM truncated conical shell. The analytical results obtained here indicate that there exist the multi-pulse Shilnikov-type homoclinic orbits for the resonant case which may result in chaos in the system. Homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are found. The diagrams show a gradual breakup of the homoclinic tree in the system as the dissipation factor is increased. Numerical simulations are presented to illustrate that for the FGM truncated conical shell, the multi-pulse Shilnikov-type chaotic motions can occur. The influence of the structural-damping, the aerodynamic-damping, and the in-plane and transverse excitations on the system dynamic behaviors is also discussed by numerical simulations. The results obtained here mean the existence of chaos in the sense of the Smale horseshoes for the FGM truncated conical shell.

Fractional dynamical behavior of electrical activity in a model of pancreatic β-cells

The electrical activity of a three dimensional fractional-order model of pancreatic β-cells is numerically studied. The dynamics of the model is understood through the Adams–Bashforth–Moulton predictor corrector scheme, which is a time-domain approach. By considering both integer-order and fractional-order models, we highlight some differences in their dynamics characteristics. It is shown that the fractional model can give rise to chaotic dynamics due to the existence of a Smale horseshoe. Some behaviors related with diabetes mellitus are also reported. The fractional model gets closer to the real biological behavior, thus provides motivation in modeling biological systems with fractional order derivatives.

Multipulse Orbits and Chaotic Dynamics of an Aero-Elastic FGP Plate Under Parametric and Primary Excitations

The multipulse global bifurcations and chaotic dynamics of a simply supported Functionally Graded Piezoelectric (FGP) rectangular plate with bonded piezoelectric layer are investigated with the case of 1:2 internal resonance and primary parametric resonance. Based on the averaged equations obtained, the theory of normal form is utilized to obtain the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues. According to the explicit expressions of normal form, the extended Melnikov method developed by Camassa et al. is employed to study the Shilnikov-type multipulse homoclinic bifurcations and chaotic dynamics of the aero-elastic FGP plate. The analytical results indicate that there exists the Shilnikov-type multipulse chaotic dynamics for the FGP plate. Numerical simulations are presented to show that for the FGP plate, the Shilnikov-type multipulse chaotic motions can occur. The influence of the in-plane excitation and the piezoelectric voltage excitation to the system dynamic behaviors is also discussed by numerical simulations. The results obtained here imply the existence of chaos in the sense of the Smale horseshoes for the FGP plate.

The Smale horseshoes in a class of non-invertible maps in with two-direction instability

We study the existence of Smale horseshoes of new type and the uniformly hyperbolic invariant sets for a class of non-invertible maps in three-dimensional Euclidean spaces with the dimension of instability equal to two. Parameter regions are given, for which the map has a horseshoe and a uniformly hyperbolic invariant set on which the map is topologically conjugate to the two-sided fullshift on four symbols.

Global dynamics of composite panels with free-layer damping treatment in subsonic flow

Global dynamics of forcedly excited composite panels with free layer damping treatment in subsonic flow near the first-order critical velocity is investigated. Hamilton’s principle is implemented to derive the PDE of such fluid-structure interaction systems. Then the governing equation is transformed into a discretized nonlinear gyroscopic system via assumed modes and Galerkin’s method. The canonical transformations and normal form theory are applied to reduce the equations of motion to near-integrable Hamiltonian standard forms considering zero to one internal resonance. The Energy-Phase method is employed to demonstrate the existence of chaotic dynamics by identifying the existence of multi-pulse jumping orbits in the perturbed phase space. In both the Hamiltonian and the dissipative perturbation case, the homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are demonstrated. In the case of dissipative perturbation, the existence of generalized Šilnikov’s type of orbits which are homoclinic to fixed points on the slow manifold are examined and the parameter region for which the dynamical system may exhibit chaotic motions in the sense of Smale horseshoes are obtained analytically. The present research illustrates that the existence of multi-pulse homoclinic orbits can provide a mechanism for how energy flow from the high-frequency mode to the low-frequency mode. The global results are finally interpreted in terms of the physical traveling wave motion of such gyroscopic continua.

Study on Chaotic Behavior of the Restricted Four-Body Problem with an Equilateral Triangle Configuration

In this paper, the chaotic behavior of a planar restricted four-body problem with an equilateral triangle configuration is analytically studied. Firstly, according to the perturbation method of Melnikov, the planar restricted four-body problem is regarded as a perturbation of the two-body model. Then, we show that the Melnikov integral function has a simple zero, arriving at the existence of transversal homoclinic orbits. Afterwards, since the standard Smale–Birkhoff homoclinic theorem cannot be directly applied to the case of a degenerate saddle, we alternatively construct an invertible map [Formula: see text] and check that [Formula: see text] is a Smale horseshoe map, showing that our restricted four-body problem possesses chaotic behavior of the Smale horseshoe type.

Global bifurcations and single‐pulse homoclinic orbits of a plate subjected to the transverse and in‐plane excitations

The Shilnikov‐type single‐pulse homoclinic orbits and chaotic dynamics of a simply supported truss core sandwich plate subjected to the transverse and the in‐plane excitations are investigated in detail. The resonant case considered here is principal parametric resonance and 1:2 internal resonance. Based on the normal form theory, the desired form for the global perturbation method is obtained. By using the global perturbation method developed by Kovacic and Wiggins, explicit sufficient conditions for the existence of a Shilnikov‐type homoclinic orbit are obtained, which implies that chaotic motions may occur for this class of truss core sandwich plate in the sense of Smale horseshoes. Numerical results obtained by using the fourth‐order Runge–Kutta method agree with theoretical analysis at least qualitatively. Copyright © 2017 John Wiley & Sons, Ltd.