Narrow Results

We analyze the global structure of the solutions of a three-dimensional, autonomous ordinary differential equation which depends on two parameters. We use graphical, heuristic, and rigorous arguments to show that as the parameters vary, a wide range of dynamical behavior is displayed.

Database Tomography (DT) is a textual database analysis system consisting of two major components: (1) algorithms for extracting multiword phrase frequencies and phrase proximities (physical closeness of the multiword technical phrases) from any type of large textual database, to augment (2) interpretative capabilities of the expert human analyst. DT was used to derive technical intelligence from a Nonlinear Dynamics database derived from the Science Citation Index/Social Science Citation Index (SCI). Phrase frequency analysis by the technical domain experts provided the pervasive technical themes of the Nonlinear Dynamics database, and the phrase proximity analysis provided the relationships among the pervasive technical themes. Bibliometric analysis of the Nonlinear Dynamics literature supplemented the DT results with author/journal/institution publication and citation data.

In this paper, bifurcations of limit cycles close to certain singularities of the vector fields are explored using an algorithm based on the harmonic balance method, the theory of nonlinear feedback systems and the monodromy matrix. Period-doubling, pitchfork and Neimark–Sacker bifurcations of cycles are detected close to a Gavrilov–Guckenheimer singularity in two modified Rössler systems. This special singularity has a zero eigenvalue and a pair of pure imaginary eigenvalues in the linearization of the flow around its equilibrium. The presented results suggest that the proposed technique can be promising in analyzing limit cycle bifurcations arising in the unfoldings of other complex singularities.

In this work a one-dimensional piecewise-smooth dynamical system, representing a Poincaré return map for dynamical systems of the Lorenz type, is investigated. The system shows a bifurcation scenario similar to the classical period-doubling one, but which is influenced by so-called border collision phenomena and denoted as border collision period-doubling bifurcation scenario. This scenario is formed by a sequence of pairs of bifurcations, whereby each pair consists of a border collision bifurcation and a pitchfork bifurcation. The mechanism leading to this scenario and its characteristic properties, like symmetry-breaking and symmetry-recovering as well as emergence of coexisting attractors, are investigated.

The appearance of infinitely-many period-doubling cascades is one of the most prominent features observed in the study of maps depending on a parameter. They are associated with chaotic behavior, since bifurcation diagrams of a map with a parameter often reveal a complicated intermingling of period-doubling cascades and chaos.

Period doubling can be studied at three levels of complexity. The first is an individual period-doubling bifurcation. The second is an infinite collection of period doublings that are connected together by periodic orbits in a pattern called a cascade. It was first described by Myrberg and later in more detail by Feigenbaum. The third involves infinitely many cascades and a parameter value μ₂ of the map at which there is chaos. We show that often virtually all (i.e. all but finitely many) "regular" periodic orbits at μ₂ are each connected to exactly one cascade by a path of regular periodic orbits; and virtually all cascades are either paired — connected to exactly one other cascade, or solitary — connected to exactly one regular periodic orbit at μ₂. The solitary cascades are robust to large perturbations. Hence the investigation of infinitely many cascades is essentially reduced to studying the regular periodic orbits of F(μ₂, ⋅). Examples discussed include the forced-damped pendulum and the double-well Duffing equation.

In this paper, we study the dynamical behavior of a prey-dependent digestive model with a state-dependent impulsive effect. Using the Poincaré map and the Lambert W-function, we find the analytical expression of discrete mapping. Sufficient conditions are established for transcritical bifurcation and period-doubling bifurcation through an analytical method. Exact locations of these bifurcations are explored. Numerical simulations of an example are illustrated which agree well with our theoretical results.

A study of a DC–DC boost converter fed by a photovoltaic (PV) generator and supplying a constant voltage load is presented. The input port of the converter is controlled using fixed frequency pulse width modulation (PWM) based on the loss-free resistor (LFR) concept whose parameter is selected with the aim to force the PV generator to work at its maximum power point. Under this control strategy, it is shown that the system can exhibit complex nonlinear behaviors for certain ranges of parameter values. First, using the nonlinear models of the converter and the PV source, the dynamics of the system are explored in terms of some of its parameters such as the proportional gain of the controller and the output DC bus voltage. To present a comprehensive approach to the overall system behavior under parameter changes, a series of bifurcation diagrams are computed from the circuit-level switched model and from a simplified model both implemented in PSIM^© software showing a remarkable agreement. These diagrams show that the first instability that takes place in the system period-1 orbit when a primary parameter is varied is a smooth period-doubling bifurcation and that the nonlinearity of the PV generator is irrelevant for predicting this phenomenon. Different bifurcation scenarios can take place for the resulting period-2 subharmonic regime depending on a secondary bifurcation parameter. The boundary between the desired period-1 orbit and subharmonic oscillation resulting from period-doubling in the parameter space is obtained by calculating the eigenvalues of the monodromy matrix of the simplified model. The results from this model have been validated with time-domain numerical simulation using the circuit-level switched model and also experimentally from a laboratory prototype. This study can help in selecting the parameter values of the circuit in order to delimit the region of period-1 operation of the converter which is of practical interest in PV systems.

We report numerical evidence of a novel chain of subharmonic resonances organized in period-adding domains in the parameter space. Different periodic domains are mediated by a coexisting main subharmonic motion according to a hierarchical organization rule. We consider the driven bilinear oscillator — which is the simplest system to model a range of engineering systems going from cracked mechanical structures to switching electronic circuits — and show that its parameter space is marked by multistability involving subharmonic main modes, as well as period-adding secondary modes which, through a period-doubling route, lead to coexisting chaotic attractors.

In this paper, a modification of homotopy analysis method (HAM) is applied to study the two-degree-of-freedom coupled Duffing system. Firstly, the process of calculating the two-degree-of-freedom coupled Duffing system is presented. Secondly, the single periodic solutions and double periodic solutions are obtained by solving the constructed nonlinear algebraic equations. Finally, comparing the periodic solutions obtained by the multi-frequency homotopy analysis method (MFHAM) and the fourth-order Runge–Kutta method, it is found that the approximate solution agrees well with the numerical solution.

In this paper, bifurcation analysis of a three-dimensional discrete game model is provided. Possible codimension-one (codim-1) and codimension-two (codim-2) bifurcations of this model and its iterations are investigated under variation of one and two parameters, respectively. For each bifurcation, normal form coefficients are calculated through reduction of the system to the associated center manifold. The bifurcations detected in this paper include transcritical, fold, flip (period-doubling), Neimark–Sacker, period-doubling Neimark–Sacker, resonance 1:2, resonance 1:3, resonance 1:4 and fold-flip bifurcations. Moreover, we depict bifurcation diagrams corresponding to each bifurcation with the aid of numerical continuation method. These bifurcation curves not only confirm our analytical results, but also reveal a richer dynamics of the model especially in the higher iterations.

In the last few decades, the discrete chaotification of difference equations has gained the considerable attention of academicians and scholars due to its tremendous applications in many branches of science, such as cryptography, traffic control models, secure communications, weather forecasting and engineering. In this article, a modulated logistic system is studied and superior chaos is reported through period-doubling bifurcation, period-three orbit and Lyapunov exponent properties. It is interesting to see that the modulated system admits the superiority in chaos due to the presence of an extra degree of freedom of an ordered parameter $\alpha$. Analytical as well as numerical simulations are demonstrated to examine the period-doubling and Lyapunov exponent properties for some particular values of parameter $\alpha$. Moreover, it is speculated that the superiority in chaos may be used in chaos-based cryptography, in the future.

In this paper, the complex dynamics of the pulsating regime of combustion waves propagation is numerically investigated within the Zel’dovich–Barenblatt–Dold model with two-step chain-branching reaction mechanism. It is shown that there exists a remarkable similarity in the dynamics of oscillations in this system and one-dimensional discrete-time maps such as the logistic map. In particular, both systems exhibit the period doubling route to chaos and the appearance of windows of stability as the bifurcation parameter is increased. For the first time for a model of combustion wave propagation the sequence of windows of stability is demonstrated and classified. Nevertheless, as the activation energy, which is taken as a bifurcation parameter, reaches a critical value corresponding to the dynamical quenching scenario the dynamics of the flame oscillations becomes multidimensional and therefore it cannot be described by a one-dimensional map anymore. The effect of the heat losses is also studied and it is demonstrated that the number of windows of stability decreases, the order of their appearance changes and the dynamical quenching appears for smaller values of the activation energy as the heat losses are increased.

This paper presents a discrete predator–prey interaction model involving hibernating vertebrates, with detailed analysis and simulation. Hibernation contributes to the survival and reproduction of organisms and species in the ecosystem as a whole. In addition, it also constitutes a wise sharing of time, space, and resources with others. We have created a new predator–prey model by integrating the two species, Holling-III and Holling-I, which have a bifurcation within a specified parameter range. We discovered that this system possesses the stability of fixed points as well as several bifurcation behaviors. To accomplish this, the center manifold theorem and bifurcation theory are applied to create existence conditions for period-doubling bifurcations and Neimark–Sacker bifurcations, which are depicted in the graph as distinct structures. Examples of numerical simulations include bifurcation diagrams, maximum Lyapunov exponents, and phase portraits, which demonstrate not just the validity of theoretical analysis but also complex dynamical behaviors and biological processes. Finally, the Ott–Grebogi–Yorke (OGY) method and phases of chaos control bifurcation were used to control the chaos of predator–prey model in hibernating vertebrates.