Smale Horseshoe Research Articles

Chaos and attraction domain of fractional Φ6‐van der Pol with time delay velocity

This article investigates the chaotic analysis and attractive domain of a fractional‐order ‐van der Pol with time delay velocity under harmonic excitation. Firstly, eight different types of bifurcation states of the system under different parameters are calculated by using the undisturbed system. Secondly, the Melnikov method is used to explore the effect of time delay velocity on the threshold of chaos in the Smale horseshoe sense under the double‐well potential and three‐well potential of the system. Finally, through numerical analysis of the phase diagram, bifurcation diagram, and maximum Lyapunov exponent, the influence of time delay velocity on system chaos is studied. The results indicate that an increase in the delay velocity coefficient will lead to the system transitioning from a chaotic state to a periodic state, while an increase in the delay velocity term will lead to the system transitioning from a periodic state to a chaotic state. In the study of system bifurcation, it is found that the position of the equilibrium points of the system changes during periodic motion. Therefore, cell mapping is used to draw the attractive domain of the system is studying the influence of initial conditions on the equilibrium point of the system and the results show that there is a close relationship between the attraction domain and the process of chaos occurrence.

Geometrical construction of a chaotic horseshoe-like map on Cantor dust [formula omitted]

Smale’s horseshoe map is one of the most important examples of 2D chaos in dynamical systems. Its chaotic behavior is often illustrated using a two-sided shift map. In this paper, a 3−dimensional horseshoe-like map is defined inspired by the geometrical motivation of the horseshoe map of Smale. Moreover, this construction is expressed on the Cantor dust (C×C×C) via ternary numbers and then the density of the periodic points is proved by using a different technique. To compute these periodic points more rapidly, we also give an algorithm. Finally, it is shown that this version of 3D horseshoe-like map is Devaney chaotic.

Nonexistence of energy function for surface diffeomorphisms with horseshoe basic sets

The Lyapunov function is a powerful tool for studying dynamical systems. In particular, the existence of such a function for any dynamical system given on compact manifold, established by C. Conley, is the basis for proving the Ω-stability criterion of dynamical systems. Moreover, the Lyapunov function, whose set of critical points of aligns with the chain-recurrent set of the system, which called an energy function, qualitatively determines the dynamics of the system. For continuous-time systems, which models dissipative processes, such a function naturally arises as an energy of the system. The existence of an energy function for arbitrary flows on compact manifolds is known. Examples of Morse–Smale diffeomorphism without energy function exist on n-dimensional manifolds starting with n>2. For systems with chaotic dynamics on surfaces, the existence of energy functions has also been established in cases where the dimension of non-trivial basic sets is greater than or equal to 1. When the chain-recurrent set of a diffeomorphism contains at least one zero-dimensional non-trivial basic set, the question of the existence of the energy function is still open. To date, it is known that the answer is negative if the zero-dimensional non-trivial basic set does not contain pairs of conjugated points. The present paper is devoted to the question of existence of an energy function for systems having a basic set with pairs of conjugated points. The primary outcome of this study is that there is no energy function for a dynamical system containing a Smale horseshoe as a basic set.

Dynamic analysis of a class of fractional‐order dry friction oscillators

This article investigates a class of Duffing nonlinear dynamic systems with fractional‐order dry friction and conducts in‐depth research on the stability, chaotic characteristics, and erosion of the safety basin of this system; the results are verified through numerical simulation. First, the average method is used to approximate the amplitude–frequency relationship of the system, and the accuracy of the analytical results is verified through numerical experiments. Second, the Melnikov method is used to obtain the conditions for the system to enter chaos in the Smale horseshoe sense, and the Melnikov curve is drawn for further verification. Then, bifurcation diagrams are drawn for the changes in various parameters in the system, with a focus on analyzing the influence of friction factors on chaotic bifurcation. By applying the definition and calculation principle of the maximum Lyapunov exponent, and drawing and utilizing the maximum Lyapunov exponent graph, the chaotic state that the system enters under different parameters is more clearly defined. Finally, the evolution law of the safety basin under various parameter changes, especially dry friction changes, is analyzed, and the erosion and bifurcation mechanism of the safety basin is studied. Comparing with the bifurcation diagram, it reveals that chaos primarily contributes to the erosion of the safety basin.

Diffeomorphisms with infinitely many Smale horseshoes

ABSTRACT A class of planar diffeomorphims is formulated, with infinitely many coexisting Smale horseshoes, where the Lebesgue measure of the parameters with such strange dynamics is infinite. On each horseshoe, there exists a uniformly hyperbolic invariant set, on which the map is topologically conjugate to the two-sided full-shift on two symbols. Moreover, the topological entropy is infinite in certain parameter regions.

Energy harvesting for a vibration isolation system with negative stiffness under harmonic excitation

ABSTRACT In this paper, an energy harvesting system with a negative stiffness structure is established, whose main purpose is to achieve good vibration isolation, and at the same time, the system can also achieve the purpose of energy harvesting. In order to analyse the system model, the amplitude–frequency response equations for the displacement, induced voltage and electric power of the system when the system is in the main resonance state are obtained by using the multiscale method, and their amplitude–frequency response curves are plotted. In addition, we obtained the existence conditions for the system to satisfy Smale's horseshoe chaos using the Melnikov function. And the phase diagram and Poincaré mapping of the system were obtained using the fourth-order Runge–Kutta method. Numerical results show that the system has some energy harvesting performance along with vibration isolation. Variations in the parameters of the system affect the resonance frequency band and co-oscillation amplitude of the system as well as the nonlinear characteristics of the system itself.

Absolutely Continuous Invariant Measure for Generalized Horseshoe Maps

Absolutely Continuous Invariant Measure for Generalized Horseshoe Maps

Chaotic dynamics of granules-beam coupled vibration: Route and threshold

Although granular materials are the second most processed in industry after water, the theoretical study of granules-structure interactions is not as advanced as that of fluid-structure interactions due to the lack of a unified view of the constitutive relation of granular materials. In the previous work [1], the theoretical model of granules-beam coupled vibration was developed and verified by experiments. However, it was also found that the system exhibits significant stiffness-softening Duffing characteristics even under micro-vibration, which implies that chaotic responses may be reached under certain conditions. To reveal the route and critical conditions for the system to enter chaos, in the present work, the chaotic dynamics of the system are studied. In qualitative analysis, Melnikov method is applied to analyze the instability behavior of the perturbed heteroclinic orbit of the system, thus the critical condition for the system to enter Smale horseshoe chaos, i.e., the Melnikov criterion, is obtained. The validity of the criterion is verified numerically. In experimental studies, the existence of chaotic responses and the route to chaos for granules-beam coupled vibrations is revealed. The experimental results suggest that the system response first experiences symmetry-breaking then undergoes a complete period-doubling cascade, and finally enters single-scroll chaos. In addition, although the Additional Dynamic Load (ADL) generated by granular media is highly complicated, a general and simple evolution pattern of the chaos threshold is found by parametric experiments, which is also supported by the Melnikov criterion. In short, the chaotic dynamics of granules-beam coupled vibration is revealed, which is a contribution to the engineering vibration aspect. On the theoretical side, an impressive result lies in that for the first time, the Melnikov criterion of such fractional-order systems is obtained in a global and closed form, which provides, respectively, improvement suggestions and reference results for the existing and future research on chaotic dynamics of fractional-order systems, especially of the Duffing-type systems.

Are almost all trajectories dense in a chaotic set?

Chaos is the most complex dynamic behavior a system can show. Yet, it is produced by two simple, though nonlinear, deterministic mechanisms, hence the more proper name Deterministic Chaos. The first mechanism, stretching, acts locally and linearly, expanding small sets in the system's state space in (at least) one direction and contracting in the others. The second mechanism, folding, acts globally and is the nonlinear operation of bending the elongated sets to form the forward image of the original sets. These two simple operations are invertible and form a chaotic set for the system's forward and backward map, containing an uncountable infinity of astonishingly complex system's trajectories. The simplest explanation of chaos I ever heard is James A. Yorke “pizza machine” (plenary lecture at the SIAM DS, Snowbird, 2003): kneading the pizza dough is nothing but stretching and folding. The mathematical formalization is Stephen Smale's horseshoe map. Using this nice tool, I revisit the hallmarks of chaos, one of which (in the title) still misses a constructive proof.

A class of non-invertible chaotic polynomial maps and Smale horseshoes of new types

A class of non-invertible chaotic polynomial maps and Smale horseshoes of new types

Generating Chaos with Saddle-Focus Homoclinic Orbit

This paper develops an anticontrol approach to design a 3D continuous-time autonomous chaotic system with saddle-focus homoclinic orbit, based on two chaotification criterions for all orbits to be globally bounded with positive Lyapunov exponents. By using the Shil’nikov theorem, a Poincaré return map near the origin is found in the designed controlled system, confirming the existence of chaos in sense of the Smale horseshoe.

Dynamical characterization of a Duffing–Holmes system containing nonlinear damping under constant excitation

A Duffing-Holmes system containing nonlinear damping is used to investigate some dynamical properties of the system under the combined action of constant excitation and simple harmonic excitation. The harmonic balance method is used to find the main vibration equation of the system and obtain the amplitude-frequency response relationship. In the analysis of the global characteristics of the system, the Melnikov method is applied to analyze the necessary, insufficient conditions for the analysis of the system chaos, and the correctness of the analytical solution is verified by numerical calculations; the effect of the constant excitation amplitude on the global characteristics of the system is analyzed by the cell mapping method. The paper shows: gradually increasing the constant excitation amplitude causes the system amplitude-frequency response to appear rigidly asymptotic phenomenon; the boundary curve of the system generating chaos in the sense of Smale's horseshoe, with the increase of excitation frequency ω, the curve first decreases, then rises, finally tends to infinity, and the possibility of chaos occurs most near the resonance frequency of the system; with the change of the constant excitation amplitude, The number of attractor domains and attractors in the global attraction domain of the system changes substantially with the change of the constant excitation amplitude, and the attractor domains are entangled with each other.

The hidden complexity of a double-scroll attractor: Analytic proofs from a piecewise-smooth system.

Double-scroll attractors are one of the pillars of modern chaos theory. However, rigorous computer-free analysis of their existence and global structure is often elusive. Here, we address this fundamental problem by constructing an analytically tractable piecewise-smooth system with a double-scroll attractor. We derive a Poincaré return map to prove the existence of the double-scroll attractor and explicitly characterize its global dynamical properties. In particular, we reveal a hidden set of countably many saddle orbits associated with infinite-period Smale horseshoes. These complex hyperbolic sets emerge from an ordered iterative process that yields sequential intersections between different horseshoes and their preimages. This novel distinctive feature differs from the classical Smale horseshoes, directly intersecting with their own preimages. Our global analysis suggests that the structure of the classical Chua attractor and other figure-eight attractors might be more complex than previously thought.

Boxing-in of a blender in a Hénon-like family

IntroductionThe extension of the Smale horseshoe construction for diffeomorphisms in the plane to those in spaces of at least dimension three may result in a hyperbolic invariant set referred to as a blender. The defining property of a blender is that it has a stable or unstable invariant manifold that appears to have a dimension larger than expected. In this study, we consider a Hénon-like family in ℝ3, which is the only explicitly given example of a system known to feature a blender in a certain range of a parameter (corresponding to an expansion or contraction rate). More specifically, as part of its hyperbolic set, this family has a pair of saddle fixed points with one-dimensional stable or unstable manifolds. When there is a blender, the closure of these manifolds cannot be avoided by one-dimensional curves coming from an appropriate direction. This property has been checked for the Hénon-like family by the method of computing extremely long pieces of global one-dimensional manifolds to determine the parameter range over which a blender exists.MethodsIn this study, we take the complimentary and local point of view of constructing an actual three-dimensional box (a parallelopiped) that acts as an outer cover of the hyperbolic set. The successive forward or backward images of this box form a nested sequence of sub-boxes that contains the hyperbolic set, as well as its respective local invariant manifold.ResultsThis constitutes a three-dimensional horseshoe that, in contrast to the idealized affine construction, is quite general and features sub-boxes with curved edges. The initial box is defined in a parameter-dependent way, and this allows us to characterize properties of the hyperbolic set intuitively.DiscussionIn particular, we trace relevant edges of sub-boxes as a function of the parameter to provide additional geometric insight into when the hyperbolic set may or may not be a blender.

Stability of nonlinear vibrations induced by rolling force in a precise cold mill system

This paper investigates the stability of horizontal nonlinear vibration of four-high cold mill under the effects of gyro-precession and eccentricity between rolls. The threshold value of chaos about Smale horseshoe commutation is given from Melnikov method, and the correctness of the result is verified by the fractal process of the attraction basin. Through the analyzes of bifurcation, maximum Lyapunov exponents and bi-parameter bifurcation, it is revealed for the effect of different parameters variation on the nonlinear dynamic behavior of horizontal vibration in rolling system. The results show, the fluctuation amplitude of horizontal force increases, the stable domain decreases, and once its value ≥3.5, it is necessary to prefer to the adjustment of parameter-exciting stiffness; meanwhile, it is noticeable that the critical points of damping and average force of the system with bifurcation or chaos appear hysteresis. The evolutions of system dynamics with four pairs of parameter changes confirm the quantitative optimal couple of one structural plus one process parameter in the system. Based on this, we can also find a balance between the average horizontal force and eccentricity, so as to optimize the structural design so that there is a proper and reliable eccentricity between the rolls. The results provide a theoretical reference for service stability and dynamic reliability of the rolling mill system.

Quantum Chaos in the Dynamics of Molecules.

Quantum chaos is reviewed from the viewpoint of "what is molecule?", particularly placing emphasis on their dynamics. Molecules are composed of heavy nuclei and light electrons, and thereby the very basic molecular theory due to Born and Oppenheimer gives a view that quantum electronic states provide potential functions working on nuclei, which in turn are often treated classically or semiclassically. Therefore, the classic study of chaos in molecular science began with those nuclear dynamics particularly about the vibrational energy randomization within a molecule. Statistical laws in probabilities and rates of chemical reactions even for small molecules of several atoms are among the chemical phenomena requiring the notion of chaos. Particularly the dynamics behind unimolecular decomposition are referred to as Intra-molecular Vibrational energy Redistribution (IVR). Semiclassical mechanics is also one of the main research fields of quantum chaos. We herein demonstrate chaos that appears only in semiclassical and full quantum dynamics. A fundamental phenomenon possibly giving birth to quantum chaos is "bifurcation and merging" of quantum wavepackets, rather than "stretching and folding" of the baker's transformation and the horseshoe map as a geometrical foundation of classical chaos. Such wavepacket bifurcation and merging are indeed experimentally measurable as we showed before in the series of studies on real-time probing of nonadiabatic chemical reactions. After tracking these aspects of molecular chaos, we will explore quantum chaos found in nonadiabatic electron wavepacket dynamics, which emerges in the realm far beyond the Born-Oppenheimer paradigm. In this class of chaos, we propose a notion of Intra-molecular Nonadiabatic Electronic Energy Redistribution (INEER), which is a consequence of the chaotic fluxes of electrons and energy within a molecule.

Bifurcation and chaos analysis of mold splicing zone vibration under thermal-mechanical coupling

To improve the machining quality of the splicing area of the hardened steel mold, this paper proposes a bifurcation and chaos analysis method for the milling vibration of the splicing area of the mold under the coupled thermal-mechanical effect. First, based on the instantaneous milling force model at the splice, the milling dynamics equations of a two-degree-of-freedom rectangular spliced mold part are established. By applying the Galerkin method, the resulting nonlinear partial differential equation is reduced to a single-degree-of-freedom nonlinear system. Meanwhile, by using the Melnikov function method, the criteria for chaos motion are obtained by the Smale horseshoe mapping. Then, the fourth-order Runge-Kutta algorithm is applied to solve the bifurcation diagram, largest Lyapunov exponent diagram, displacement waveform diagram, and phase plane diagram of the nonlinear dynamic system in the mold splicing area. Subsequently, the influence of spindle speed and cutting depth on the nonlinear vibration system in the mold splice area is quantitatively analyzed. Finally, the effect of spindle speed and cutting depth on the vibration system of the workpiece is investigated through experiments. The obtained results indicate that the milling vibration amplitude in the rectangular splicing mold area can be controlled more easily and effectively by changing the cutting depth than by changing the spindle speed.

Chaos of the Six-Dimensional Non-Autonomous System for the Circular Mesh Antenna

In the process of aerospace service, circular mesh antennas generate large nonlinear vibrations under an alternating thermal load. In this paper, the Smale horseshoe and Shilnikov-type multi-pulse chaotic motions of the six-dimensional non-autonomous system for circular mesh antennas are first investigated. The Poincare map is generalized and applied to the six-dimensional non-autonomous system to analyze the existence of Smale horseshoe chaos. Based on the topological horseshoe theory, the three-dimensional solid torus structure is mapped into a logarithmic spiral structure, and the original structure appears to expand in two directions and contract in one direction. There exists chaos in the sense of a Smale horseshoe. The nonlinear equations of the circular mesh antenna under the conditions of the unperturbed and perturbed situations are analyzed, respectively. For the perturbation analysis of the six-dimensional non-autonomous system, the energy difference function is calculated. The transverse zero point of the energy difference function satisfies the non-degenerate conditions, which indicates that the system exists Shilnikov-type multi-pulse chaotic motions. In summary, the researches have verified the existence of chaotic motion in the six-dimensional non-autonomous system for the circular mesh antenna.

A simple topological model for two coupled neurons.

A simple topological model describing the chaotic dynamics of two coupled neurons is established and analyzed based on the Smale horseshoe theory.

Boundary condition induced passive chaotic mixing in straight microchannels

When fluids flow through straight channels sustained turbulence occurs only at high Reynolds numbers [typically Re∼O(1000)]. It is difficult to mix multiple fluids flowing through a straight channel in the low Reynolds number laminar regime [Re<O(100)] because in the absence of turbulence, mixing between the component fluids occurs primarily via the slow molecular diffusion process. This Letter reports a simple way to significantly enhance the low Reynolds number (in our case Re≤10) passive microfluidic flow mixing in a straight microchannel by introducing asymmetric wetting boundary conditions on the floor of the channel. We show experimentally and numerically that by creating carefully chosen two-dimensional hydrophobic slip patterns on the floor of the channels, we can introduce stretching, folding, and/or recirculation in the flowing fluid volume, the essential elements to achieve mixing in the absence of turbulence. We also show that there are two distinctive pathways to produce homogeneous mixing in microchannels induced by the inhomogeneity of the boundary conditions. It can be achieved either by (1) introducing stretching, folding and twisting of fluid volumes, i.e., via a horse-shoe type transformation map, or (2) by creating chaotic advection, achieved through manipulation of the hydrophobic boundary patterns on the floor of the channels. We have also shown that by superposing stretching and folding with chaotic advection, mixing can be optimized in terms of significantly reducing mixing length, thereby opening up new design opportunities for simple yet efficient passive microfluidic reactors.