Narrow Results

The alteration in concentration of atmospheric pollutants is influencing the functionality and growth of forests. Also, the growing human demand for forestry resources is detrimentally affecting the sustainability of these valuable resources. In this study, we present a mathematical model that incorporates the influence of atmospheric pollutants on the intrinsic growth rate of forests, while concurrently addressing the utilization of forestry land by the human population for diverse purposes which diminishes the carrying capacity of forestry resources. We establish sufficient conditions under which all relevant dynamic variables stabilize at their equilibria. Upon scrutinizing the model system, we observe multiple bifurcations concerning certain key parameters. Additionally, numerical simulations have been conducted to corroborate the analytically derived findings. Moreover, we fortify the proposed model through the integration of a time delay in the impact of pollutants on the intrinsic growth rate of forestry resources. Despite the conventional belief that introducing a time delay tends to destabilize systems, our resolute delayed model system showcases that a time delay in the effect of pollutants on intrinsic growth rate of forestry resources can, in fact, stabilize the unstable interior equilibrium.

In this paper, we use the techniques from dynamical systems and singular traveling wave theory developed by [Li & Chen, 2007] to investigate the exact explicit solutions for the perturbed Gerdjikov–Ivanov (GI) equation. By considering the corresponding dynamical system and finding the bifurcations of phase portraits for the amplitude component of the traveling solutions, the dynamical behavior can be revealed. Under different parameter conditions, exact explicit solutions of the perturbed GI equation are found.

Analyzing tumor growth dynamics improves treatment strategies for cancer. Many models have been proposed to analyze cancer development dynamics, which do not always exhibit chaotic behavior. This research aims to create and examine a unique dynamical cancer model that exhibits chaotic behavior for certain parameters. Introducing chaos into the model allows for the exploration of irregular tumor growth patterns and the identification of critical thresholds that can influence treatment outcomes. The model is examined, and each system parameter’s impact on the model’s dynamics is evaluated. The analysis of the bifurcation and Lyapunov diagrams demonstrates chaos in three populations of tumor, healthy, and immune system cells. By suppressing the immunological response, the cancer cell gains control of the chaotic attractor and establishes a stable state. This might lower the cancer condition by altering the appropriate parameter range assisting in tumor treatment.

In this paper, we consider a class of near-Hamiltonian systems with a nilpotent center, and study the number of limit cycles including algebraic limit cycles. We prove that there are at most $n(m+1)+1$ large amplitude limit cycles if the first-order Melnikov function is not zero identically, including an algebraic limit cycle. Moreover, it can have $\frac{n(n+3)}{2}$ when $m\ge n$ and $\frac{m(2n-m+1)}{2}+n$ when $m<n$ small limit cycles. We also provide two examples as applications of our main results.

The paper is devoted to analyzing the mechanisms of spread and prevention of epidemics, based on a discrete model that takes into account the infection spread because of contacts among infected and susceptible individuals, disease-induced mortality, and the factor of treatment. Pathways leading to the complete extinction, complete recovery, and nontrivial coexistence of susceptible and infected individuals are revealed by bifurcation analysis. Parametric conditions of nontrivial modes of coexistence in the form of equilibrium, discrete cycle, quasiperiodic closed invariant curve and chaos are found. An extended analysis on transformation scenarios of these regimes in dependence of the variation of the rate of the infection spread and the treatment intensity is performed. Phenomena of infection-induced chaos and its suppression by treatment are discussed.

We investigate the phase transition between solutions with distinct symmetrical property observed in a system of coupled three-oscillators with hard characteristics and state coupling. By a symmetry-breaking bifurcation, a symmetrical in-phase solution bifurcates into synchronized modes with a partially in-phase solution and an almost in-phase solution. Moreover, by using the definition of symmetrical and asymmetrical three-phase solutions, we confirmed the existence of a stable symmetrical three-phase quasi-periodic solution and an asymmetrical three-phase chaotic solution in the coupled system.

The Josephson equation with constant current and sinusoidal forcings and a phase shift is investigated in detail: the existence and the bifurcations of harmonics and subharmonics under small perturbations are given, by using the second-order averaging method and Melnikov function; the influence on bifurcations of periodic or subharmonics as the phase shift varies is considered; some numerical simulation results are reported in order to prove our theoretical predictions.

We present a robust method to locate and continue period-doubling, saddle-node and symmetry-breaking bifurcations of periodically driven experimental systems. The method is illustrated from results obtained for an electronic implementation of a Duffing oscillator.

The dynamics of a network of three neurons with all possible connections is studied here. The equations of control are given by three differential equations with nonlinear, positive and bounded sigmoidal response function of the neurons. The system passes from stable to periodic and then to chaotic regimes and returns to stationary regime with change in parameter values of synaptic weights and decay rates. We have developed programs and used Locbif package to study phase portraits, bifurcation diagrams which confirm the result. Lyapunov Exponents have been calculated to confirm chaos.

This paper investigates the complex dynamics, synchronization and control of chaos in a system of strongly connected Wilson–Cowan neural oscillators. Some typical synchronized periodic solutions are analyzed by using the Poincaré mapping method, for which bifurcation diagrams are obtained. It is shown that topological change of the synchronization mode is mainly caused and carried out by the Neimark–Sacker bifurcation. Finally, a simple feedback control method is presented for stabilizing an in-phase synchronizing periodic solution embedded in the chaotic attractor of a higher-dimensional model of such coupled neural oscillators.

Using the method of qualitative analysis we show that five perturbed cubic Hamiltonian systems have the same distribution of limit cycles and have 11 limit cycles for some parameters. The accurate location of each limit cycle is given by numerical exploration. In other words, we demonstrate the existence of 11 limit cycles and their distribution in five perturbed systems in two ways, the results obtained from both ways are the same.

In this paper, we perform our algorithm developed in [Yu & Lee, 2001] to present the entire branches of quasiperiodic solutions starting from the bifurcation points in the branches of periodic solutions in an interval of parameters for the 2-mode damped, driven sine-Gordon ODE.

It is well known that a nonorientable manifold in a three-dimensional vector field is topologically equivalent to a Möbius strip. The most frequently used example is the unstable manifold of a periodic orbit that just lost its stability in a period-doubling bifurcation. However, there are not many explicit studies in the literature in the context of dynamical systems, and so far only qualitative sketches could be given as illustrations. We give an overview of the possible bifurcations in three-dimensional vector fields that create nonorientable manifolds. We mainly focus on nonorientable manifolds of periodic orbits, because they are the key building blocks. This is illustrated with invariant manifolds of three-dimensional vector fields that arise from applications. These manifolds were computed with a new algorithm for computing two-dimensional manifolds.

The dynamical systems of planet–belt interaction are studied by the fixed-point analysis, and the bifurcation of solutions on the parameter space is discussed. For most cases, our analytical and numerical results show that the locations of fixed points are determined by the parameters and these fixed points are either structurally stable or unstable. In addition to that, there are two special fixed points: The one on the inner edge of the belt is asymptotically stable and the one on the outer edge of the belt is unstable. This is consistent with the observational picture of Asteroid Belt between Mars and Jupiter: Mars moves steadily close to the inner edge but Jupiter is quite far from the outer edge.

Two-degree-of-freedom autonomous system with friction is analyzed numerically. The friction coefficient has been smoothened using arc tan function. The standard, but slightly modified chaos identification tools have been applied for the analyzed discontinuous system. Some interesting examples of stick-slip regular and chaotic dynamics have been illustrated and discussed.

Local bifurcation control designs have been addressed in the literature for stationary, Hopf, and period doubling bifurcations. This paper addresses the local feedback control of the Neimark–Sacker bifurcation, in which an invariant closed curve emerges from a nominal fixed point of a discrete-time system as a parameter is slowly varied. The analysis of this bifurcation is more involved than for previously considered bifurcations. The paper develops the stability and amplitude equations for the bifurcated invariant curves of the Neimark–Sacker bifurcation, and then proceeds to apply these relationships in the design of nonlinear feedbacks. The feedback controllers are applied to two examples: the delayed logistic map and a model reference adaptive control system model.

We study numerical methods for solving nonlinear elliptic eigenvalue problems which contain folds and bifurcation points. First we present some convergence theory for the MINRES, a variant of the Lanczos method. A multigrid-Lanczos method is then proposed for tracking solution branches of associated discrete problems and detecting singular points along solution branches. The proposed algorithm has the advantage of being robust and can be easily implemented. It can be regarded as a generalization and an improvement of the continuation-Lanczos algorithm. Our numerical results show the efficiency of this algorithm.

BVP oscillator is the simplest mathematical model describing dynamical behavior of neural activity. Large scale neural network can often be described naturally by coupled systems of BVP oscillators. However, even if two BVP oscillators are merely coupled by a linear element, the whole system exhibits complicated behavior. In this letter, we analyze coupled BVP oscillators with asymmetrical coupling structure, besides, each oscillator has different internal resistance. The system shows a rich variety of bifurcation phenomena and strange attractors. We calculate bifurcation diagrams in two-parameter plane around which the chaotic attractors mainly appear and confirm relaxant phenomena in the laboratory experiments. We also briefly report a conspicuous strange attractor.

A plot of frequency, or spectral, content versus a system parameter was introduced in a recent paper by Billings and Boaghe [2001] as a useful alternative to bifurcation diagrams in nonlinear dynamics. The current contribution illustrates the same approach based on data taken from two experimental mechanical systems in which hysteresis is featured.

This paper presents some analytical criteria for local activity principle in reaction–diffusion Cellular Nonlinear Network (CNN) cells [Chua, 1997, 1999] that have four local state variables with three ports. As a first application, a cellular nonlinear network model of tumor growth and immune surveillance against cancer (GISAC) is discussed, which has cells defined by the Lefever–Erneaux equations, representing the densities of alive and dead cancer cells, as well as the number of free and bound cytotoxic cells, per unit volume. Bifurcation diagrams of the GISAC CNN provide possible explanations for the mechanism of cancer diffusion, control, and elimination. Numerical simulations show that oscillatory patterns and convergent patterns (representing cancer diffusion and elimination, respectively) may emerge if selected cell parameters are located nearby or on the edge of the chaos domain. As a second application, a smoothed Chua's oscillator circuit (SCC) CNN with three ports is studied, for which the original prototype was introduced by Chua as a dual-layer two-dimensional reaction–diffusion CNN with three state variables and two ports. Bifurcation diagrams of the SCC CNN are computed, which only demonstrate active unstable domains and edges of chaos. Numerical simulations show that evolution of patterns of the state variables of the SCC CNN can exhibit divergence, periodicity, and chaos; and the second and the fourth state variables of the SCC CNNs may exhibit generalized synchronization. These results demonstrate once again Chua's assertion that a wide spectrum of complex behaviors may exist if the corresponding CNN cell parameters are chosen in or nearby the edge of chaos.