International Journal of Bifurcation and Chaos

This paper provides a theoretical background for Lagrangian Descriptors (LDs). The goal of achieving rigorous proofs that justify the ability of LDs to detect invariant manifolds is simplified by introducing an alternative definition for LDs. The definition is stated for $n$-dimensional systems with general time dependence, however we rigorously prove that this method reveals the stable and unstable manifolds of hyperbolic points in four particular 2D cases: a hyperbolic saddle point for linear autonomous systems, a hyperbolic saddle point for nonlinear autonomous systems, a hyperbolic saddle point for linear nonautonomous systems and a hyperbolic saddle point for nonlinear nonautonomous systems. We also discuss further rigorous results which show the ability of LDs to highlight additional invariants sets, such as $n$-tori. These results are just a simple extension of the ergodic partition theory which we illustrate by applying this methodology to well-known examples, such as the planar field of the harmonic oscillator and the 3D ABC flow. Finally, we provide a thorough discussion on the requirement of the objectivity (frame-invariance) property for tools designed to reveal phase space structures and their implications for Lagrangian descriptors.

Control-based continuation (CBC) is a means of applying numerical continuation directly to a physical experiment for bifurcation analysis without the use of a mathematical model. CBC enables the detection and tracking of bifurcations directly, without the need for a post-processing stage as is often the case for more traditional experimental approaches. In this paper, we use CBC to directly locate limit-point bifurcations of a periodically forced oscillator and track them as forcing parameters are varied. Backbone curves, which capture the overall frequency-amplitude dependence of the system’s forced response, are also traced out directly. The proposed method is demonstrated on a single-degree-of-freedom mechanical system with a nonlinear stiffness characteristic. Results are presented for two configurations of the nonlinearity — one where it exhibits a hardening stiffness characteristic and one where it exhibits softening-hardening.

This paper presents examples of hysteresis from a broad range of scientific disciplines and demonstrates a variety of forms including clockwise, counterclockwise, butterfly, pinched and kiss-and-go, respectively. These examples include mechanical systems made up of springs and dampers which have been the main components of muscle models for nearly one hundred years. For the first time, as far as the authors are aware, hysteresis is demonstrated in single fibre muscle when subjected to both lengthening and shortening periodic contractions. The hysteresis observed in the experiments is of two forms. Without any relaxation at the end of lengthening or shortening, the hysteresis loop is a convex clockwise loop, whereas a concave clockwise hysteresis loop (labeled as kiss-and-go) is formed when the muscle is relaxed at the end of lengthening and shortening. This paper also presents a mathematical model which reproduces the hysteresis curves in the same form as the experimental data.

There is a continuous demand on novel chaotic generators to be employed in various modeling and pseudo-random number generation applications. This paper proposes a new chaotic map which is a general form for one-dimensional discrete-time maps employing the power function with the tent and logistic maps as special cases. The proposed map uses extra parameters to provide responses that fit multiple applications for which conventional maps were not enough. The proposed generalization covers also maps whose iterative relations are not based on polynomials, i.e. with fractional powers. We introduce a framework for analyzing the proposed map mathematically and predicting its behavior for various combinations of its parameters. In addition, we present and explain the transition map which results in intermediate responses as the parameters vary from their values corresponding to tent map to those corresponding to logistic map case. We study the properties of the proposed map including graph of the map equation, general bifurcation diagram and its key-points, output sequences, and maximum Lyapunov exponent. We present further explorations such as effects of scaling, system response with respect to the new parameters, and operating ranges other than transition region. Finally, a stream cipher system based on the generalized transition map validates its utility for image encryption applications. The system allows the construction of more efficient encryption keys which enhances its sensitivity and other cryptographic properties.

Laser-jet induced wet etching of stainless steel in 5M H₃PO₄ has been investigated. By this method, it is possible to cut and microstructure metals and alloys that form passive layers in strong etchants. Due to the laser heating of the metal and the adjacent layers of the etchant, the passive layer is removed and an active dissolution of the base metal together with the formation of hydrogen is observed. The reactions are limited by the transport of fresh acid and the removal of dissolved metal. A jet of etchant reduces the transport limitations. For definite ranges of the laser power, the feed velocity and the etchant jet velocity, a regime of periodic structure formation of the kerf, often called ripples, has been found. The ripple length depends on all three parameters. The ripple formation can be brought into correlation with a periodic change of the intensity of the reflected light as well as oscillations of the potential workpiece. It could be shown that the periodic structure formation is connected to a spreading of an etching front from the laser activated area, that temporarily moves ahead to the laser. This leads to modulations of the interface for the laser absorption, which results, for example, in oscillations of the intensity of the reflected light. This means the laser induced etching reaction attracts a feedback based on the conditions of absorption for the laser. For those parameters of feed velocity, laser power and etchant jet velocity, without ripple formation the laser induced etching front is of a constant distance to the laser which results in steady conditions at the interface for the absorption of the laser.

This paper is aimed at describing the whole set of necessary and sufficient conditions for the emergence of multiple equilibria and global indeterminacy in the standard endogenous growth framework with learning by doing. The novelty of this paper relies on the application of the original Bogdanov–Takens bifurcation theorem, which allows us to characterize the full dynamics of the model, and determine the emergence of an unavoidable poverty trap.

We investigate the stability of Turing-type patterns in one spatial dimension in a system of reaction–diffusion equations with a term depending linearly on the spatial position. The system is a generic model of two interacting chemical species where production rates are dependent on a linear external gradient. This is motivated by mathematical models in developmental biology. In a previous paper, we found analytic approximations of Turing-like steady state patterns. In the present article, we derive conditions for the stability of these patterns and show bifurcation diagrams in two small parameters related to the slope of the external gradient and the ratio of the diffusion coefficients.

The application of synchronization theory to build new cryptosystems has been a hot topic during the last two decades. In this paper, we analyze a recent proposal in this field. We pinpoint the main limitations of the software implementation of chaos-based systems designed on the grounds of synchronization theory. In addition, we show that the cryptosystem under evaluation possesses serious security problems that imply a clear reduction of the key space.

We complete the study of the bifurcations of saddle/spiral bimodal linear systems, depending on the respective traces $T$ and $\tau$: one 2-codimensional bifurcation; four kinds of 1-codimensional bifurcations. We stratify the bifurcation set in the $(T,\tau )$-plane and we describe the qualitative changes of the dynamical behavior at each bifurcation point.

We introduce a simple and efficient experimental setup for the Malkus–Lorenz waterwheel. Through a series of image processing techniques, our work is listed as one of the few experiments that measure not only the angular velocity but also the mass distribution. Our experiment is to observe qualitative changes on the waterwheel as the leakage rate changes, while the other physical parameters are fixed. We perform a bifurcation analysis for the qualitative changes, and the phase portraits from experiments are validated by the bifurcation analysis.

We mainly investigate the existence, stability and number of invariant cones in 3-dim homogeneous piecewise linear systems with two zones separated by a plane containing the 1-dim invariant manifold of each linear subsystem. By transforming the system into a proper form with the 1-dim invariant manifolds on the separation plane either coincident or perpendicular, we obtain complete results on the existence, stability and number of invariant cones and show that the maximum number of invariant cones is two. The explicit parameter relations obtained here contribute to understanding and investigating bifurcation phenomena occurring in nonsmooth dynamical systems.

Interaction between dynamical systems has been a subject of intensive study for the past couple of decades. These studies have mainly focused on synchronization of chaotic systems, conditions of different kinds of synchronized behavior, amplitude death, etc. Synchronization of periodic oscillators and the frequency of the resulting synchronized behavior have remained relatively unexplored. In this paper we consider synchronization of nonidentical periodic oscillators for different coupling schemes, and study the nature of the synchronized frequency. Based on numerical and experimental observations we show that for directly coupled oscillators, the synchronized frequency lies between the individual frequencies and its value does not depend on the coupling constant, while for indirectly coupled oscillators the synchronized frequency lies out of the range and depends on the strength of coupling. We explain the different frequency behaviors of directly and indirectly coupled systems by analytically deriving the expressions of synchronized frequency under certain simplifying assumptions.

In this paper, we investigate the multiple stick-slip chaotic motion of an archetypal self-excited smooth and discontinuous (SD) oscillator driven by moving belt friction, which is constructed with the SD oscillator and the classical moving belt. The friction force between the mass and the belt is modeled as a Coulomb friction for this system. The energy introduction or dissipation during the slip and stick modes in the system is analyzed. The analytical expressions of homoclinic orbits of the unperturbed SD oscillator are derived by using a special coordinate transformation without any pronominal truncation to retain the natural characteristics, which allows us to utilize the Melnikov’s method to obtain the chaotic thresholds of the self-excited SD oscillator in the presence of the damping and external excitation. Numerical simulations are carried out to demonstrate the multiple stick-slip dynamics of the system, which show the efficiency of the prediction for stick-slip chaos of the perturbed self-excited system. The results presented herein this paper demonstrate the complicated dynamics of stick-slip periodic solutions, multiple stick-slip chaotic solutions and also coexistence of multiple solutions for the perturbed self-excited SD oscillator.

Many types of neurons and excitable cells could intrinsically generate bursting activity, even in an isolated case, which plays a vital role in neuronal signaling and synaptic plasticity. In this paper, we have mainly investigated bursting types and corresponding bifurcations in the pre-Bötzinger complex respiratory rhythm neuron by using fast–slow dynamical analysis. The numerical simulation results have showed that for some appropriate parameters, the neuron model could exhibit four distinct types of fast–slow bursters. We also explored the bifurcation mechanisms related to these four types of bursters through the analysis of phase plane. Moreover, the first Lyapunov coefficient of the Hopf bifurcation, which can decide whether it is supercritical or subcritical, was calculated with the aid of MAPLE software. In addition, we analyzed the codimension-two bifurcation for equilibria of the whole system and gave a detailed theoretical derivation of the Bogdanov–Takens bifurcation. Finally, we obtained expressions for a fold bifurcation curve, a nondegenerate Hopf bifurcation curve, and a saddle homoclinic bifurcation curve near the Bogdanov–Takens bifurcation point.

This paper focuses on a three-species Lotka–Volterra food chain model with cross-diffusion under homogeneous Neumann boundary conditions. The known results indicate that no spatiotemporal patterns happen in the corresponding reaction–diffusion system. When some cross-diffusion terms are introduced in the system, the existence of nonconstant positive steady-states as well as the Hopf bifurcation is studied. Our result shows that cross-diffusion plays a crucial role in the formation of spatiotemporal patterns, that is, it can create not only stationary patterns but also spatially inhomogeneous periodic oscillatory patterns, which is a strong contrast to the case without cross-diffusion.

In this paper, we consider the exact explicit solutions for the famous generalized Hénon–Heiles (H–H) system. Corresponding to the three integrable cases, on the basis of the investigation of the dynamical behavior and level curves of the planar dynamical systems, we find all possible explicit exact parametric representations of solutions in the invariant manifolds of equilibrium points in the four-dimensional phase space. These solutions contain quasi-periodic solutions, homoclinic solutions, periodic solutions as well as blow-up solutions. Therefore, we answer the question: what are the flows in the center manifolds and homoclinic manifolds of the generalized Hénon–Heiles (H–H) system. As an application of the above results, we consider the traveling wave solutions for the coupled $(n+1)$-dimensional Klein–Gordon–Schrödinger Equations with quadratic power nonlinearity.

A drillbit–rock vibro-impact system with a linear rock contact law is considered in this paper. This system is a simplified model to study the drillbit oscillation in a percussive drilling system. The main purpose of this work is to explore the optimized vibration condition of the drillbit to improve the drilling efficiency in practical drilling applications. In order to eliminate the influences of drilling progression on the dynamic analysis of drillbit oscillation, the drillbit oscillation is separated from its drift via oscillation reconstruction. The obtained bounded mathematical model is analyzed by numerical integration and numerical continuation. Based on the codimension-one bifurcations detected by TC-HAT during one-parameter continuation, the two-parameter bifurcation curves are traced by two-parameter continuation. These explored two-parameter bifurcation curves constitute the parameter boundaries for the period-one one-impact oscillation of drillbit, which is verified as the optimal oscillation with the highest drilling efficiency.

Designing a pseudorandom number generator (PRNG) is a difficult and complex task. Many recent works have considered chaotic functions as the basis of built PRNGs: the quality of the output would indeed be an obvious consequence of some chaos properties. However, there is no direct reasoning that goes from chaotic functions to uniform distribution of the output. Moreover, embedding such kind of functions into a PRNG does not necessarily allow to get a chaotic output, which could be required for simulating some chaotic behaviors.

In a previous work, some of the authors have proposed the idea of walking into a $N$-cube where a balanced Hamiltonian cycle has been removed as the basis of a chaotic PRNG. In this article, all the difficult issues observed in the previous work have been tackled. The chaotic behavior of the whole PRNG is proven. The construction of the balanced Hamiltonian cycle is theoretically and practically solved. An upper bound of the expected length of the walk to obtain a uniform distribution is calculated. Finally practical experiments show that the generators successfully pass the classical statistical tests.

In this paper, we are concerned with the unilateral global bifurcation structure of fractional differential equation

We introduce a mathematical model on the dynamics of demand and supply incorporating collectability and saturation factors. Our analysis shows that when the fluctuation of the determinants of demand and supply is strong enough, there is chaos in the demand-supply dynamics. Our numerical simulation shows that such a chaos is not an attractor (i.e. dynamics is not approaching the chaos), instead a periodic attractor (of period-3 under the Poincaré period map) exists near the chaos, and coexists with another periodic attractor (of period-1 under the Poincaré period map) near the market equilibrium. Outside the basins of attraction of the two periodic attractors, the dynamics approaches infinity indicating market irrational exuberance or flash crash. The period-3 attractor represents the product’s market cycle of growth and recession, while period-1 attractor near the market equilibrium represents the regular fluctuation of the product’s market. Thus our model captures more market phenomena besides Marshall’s market equilibrium. When the fluctuation of the determinants of demand and supply is strong enough, a three leaf danger zone exists where the basins of attraction of all attractors intertwine and fractal basin boundaries are formed. Small perturbations in the danger zone can lead to very different attractors. That is, small perturbations in the danger zone can cause the market to experience oscillation near market equilibrium, large growth and recession cycle, and irrational exuberance or flash crash.