Alternative stable states and limit cycles in a three-species ecological system

Diversity in interaction strength promotes rich dynamical behaviours in a three-species ecological system

This paper reports the finding of unusual hidden and self-excited coexisting dynamical behaviors in an existing Lorenz-like system. For different parameters, the system has different types of equilibrium points, such as saddle-nodes, stable focus-nodes, saddle-foci and nonhyperbolic equilibrium points, which can be used to find different types of hidden and self-excited attractors. The different types of attractors have been vividly demonstrated by several numerical techniques including phase portraits, bifurcation diagrams and basins of attraction. Very interestingly, we find the rare coexistence of chaotic attractor and periodic orbits in the Lorenz-like system with two saddle-foci.

The interplay of rock-paper-scissors competition and environments mediates species coexistence and intriguing dynamics

The study of bursting oscillations induced by frequency-domain multiscale effect is one of the key scientific issues in the theoretical analysis of circuit systems. In this paper, we explore the mechanism of the bursting oscillations of a van der Pol-Duffing-Jerk circuit oscillator with slow-changing parametric and external periodic excitations. Three typical bursting modes, namely, left-right symmetric ‘subHopf/fold limit cycle’ bursting, origin symmetric ‘fold/fold limit cycle’ bursting and origin symmetric ‘fold/subHopf/fold limit cycle’ bursting, are presented. The slowly changing excitation is treated as a generalized state variable to analyze the influence on the critical manifolds of the equilibria and bifurcations. The critical conditions of fold and Hopf bifurcations are computed by using the bifurcation theory, and two typical bifurcation structures are obtained according to the position of different bifurcation curves. Based on the bifurcation analysis, we investigate the appearance and dynamicalal evolutions of the different bursting oscillations with the variation of the external excitation amplitude. It is pointed that not only the bifurcation structures but also the distance between the fold and Hopf bifurcation points can affect the bursting patterns. We find the directions of the trajectories and the bursting types are sensitive to the values of the external excitation amplitude. Furthermore, we reveal the mechanism of the bursting oscillations by overlapping the trajectories on (θ, x)-plane onto the corresponding bifurcation structures. Numerical simulations are also presented to prove the correctness of the theoretical analysis in our study.

Conventional multi-scroll chaotic systems (MSCSs) typically exhibit uniform scroll distributions, limiting the diversity of attractor structures. In contrast, non-uniformly distributed MSCSs can overcome this constraint, which enables more flexible attractor configurations and enhances their potential in practical engineering applications. In this study, five modified sawtooth wave functions are proposed and embedded into a three-dimensional chaotic system to generate five types of multi-scroll attractors with irregular spatial distributions, including (1) attractors with enlarged scroll structures on both sides, (2) attractors with an enlarged central scroll structure, (3) attractors with a central separation structure, (4) attractors with enlarged scroll structures at the center and both sides, and (5) attractors with separated scrolls and enlarged side scrolls. Among these, the third and fifth types exhibit attractor coexistence. Furthermore, by selecting and combining two different modified sawtooth functions, four types of grid multi-scroll attractors are constructed: (1) attractors with separated structures and varying scroll sizes, (2) attractors with cross-shaped separated structures, (3) attractors with a double-chain structure, and (4) attractors with a triple-chain structure. Among them, the cross-shaped type also exhibits attractor coexistence. This study systematically analyses the generation mechanisms of these non-uniform multi-scroll attractors and examines their offset-boosting phenomenon. The chaotic characteristics of different types of attractors are analyzed using the largest Lyapunov exponent, bifurcation diagrams, and spectral entropy. In addition, the National Institute of Standards and Technology test is employed to validate the randomness of the proposed systems. Finally, hardware implementation on a digital signal processing platform confirms its applicability for practical engineering applications.

Hysteresis bifurcation and application to delayed FitzHugh-Nagumo neural systems

Reflexes are important in the control of such daily activities as standing and walking. The goal of this study is to establish how reflexive feedback of muscle length, velocity, and force can lead to stable equilibria (i.e., posture) and limit cycles (e.g., ankle clonus and gait). The influence of stretch reflexes on the behavior and stability of musculoskeletal systems was examined using a model of human stance. We computed branches of fold and Hopf bifurcations by numerical bifurcation analysis of the model. These fold and Hopf branches divide the parameter space, constructed by the reflexive feedback gains, into regions of different behavior: unstable posture, stable posture, and stable limit cycles. These limit cycles correspond to a neural deficiency, termed ankle clonus. We also linked bifurcation analysis to known biomechanical concepts by linearizing the model: the fold branch corresponds to zero ankle stiffness and defines the minimal muscle length feedback necessary for stable posture; the Hopf branch is related to unstable reflex loops. Crossing the Hopf branch can lead to the above-mentioned stable limit cycles. The Hopf branch reduces with increasing time delays, making the subject's posture more susceptible to unstable reflex loops. This might be one of the reasons why elderly people, or those with injuries to the central nervous system, often have trouble with standing and other posture tasks. The influence of cocontraction and force feedback on the behavior of the posture model was also investigated. An increase in cocontraction leads to an increase in ankle stiffness (i.e., intrinsic muscle stiffness) and a decrease in the effective reflex loop gain. On the one hand, positive force feedback increases the ankle stiffness (i.e., intrinsic and reflexive muscle stiffness); on the other hand it makes the posture more susceptible to unstable reflex loops. For negative force feedback, the opposite is true. Finally, we calculated areas of reflex gains for perturbed stance and quiet stance in healthy subjects by fitting the model to data from the literature. The overlap of these areas of reflex gains could indicate that stretch reflexes are the major control mechanisms in both quiet and perturbed stance. In conclusion, this study has successfully combined bifurcation analysis with the more common biomechanical concepts and tools to determine the influence of reflexes on the stability and quality of stance. In the future, we will develop this line of research to look at rhythmic tasks, such as walking.

We study the evolution of a conservative dynamical system with three degrees of freedom, where small nonconservative terms are added. The conservative part is a Hamiltonian system, describing the motion of a planetary system consisting of a star, with a large mass, and of two planets, with small but not negligible masses, that interact gravitationally. This is a special case of the three body problem, which is nonintegrable. We show that the evolution of the system follows the topology of the conservative part. This topology is critically determined by the families of periodic orbits and their stability. The evolution of the complete system follows the families of the conservative part and is finally trapped in the resonant orbits of the Hamiltonian system, in different types of attractors: chaotic attractors, limit cycles or fixed points.

The Chen system of equations exhibits Lorenz, Transition, Chen and Transverse 8 type of chaotic attractors depending on the system parameters. Some authors have proposed a generalized competitive mode (GCM) technique to explain the topological difference between the Lorenz attractor and the Chen attractor. In this paper, we show a range of parameter values for which the nature of the topological attractor for the Chen system is not in accordance with that expected from GCM analysis. Instead, we find that return maps can be used to characterize the transition between different types of attractors more reliably.

Strange attractors can be observed in chaotic and hyperchaotic systems. Most of the dynamical systems hold a finite number of attractors, while some chaotic systems can be controlled to present an infinite number of attractors by generating infinite equilibria. Chaos can also be triggered in some dynamical systems that can present hidden attractors, and the attractors in these dynamical systems find no equilibria and the basin of attraction is not connected with any equilibrium (the equilibria position meets certain restriction function). In this paper, Hamilton energy is calculated on the chaotic systems with different types of attractors, and energy modulation is used to control the chaos in these systems. The potential mechanism could be that negative feedback in energy can suppress the phase space and oscillating behaviors, and thus, the chaotic, periodical oscillators can be controlled. It could be effective to control other chaotic, hyperchaotic and even periodical oscillating systems as well.

The interplay between mutualism, competition and dispersal promotes species coexistence in a multiple interactions type system

The behavior of a nonlinear, non-Hamiltonian system in the postcritical (flutter) domain is studied with special attention to the influence of initial conditions on the properties of attractors situated at a certain point of the control parameter space. As a prototype system, an elastic panel is considered that is subjected to a combination of supersonic gas flow and quasistatic loading in the middle surface. A two natural modes approximation, resulting in a four-dimensional phase space and several control parameters is considered in detail. For two fixed points in the control parameter space, several plane sections of the four-dimensional space of initial conditions are presented and the asymptotic behavior of the final stationary responses are identified. Amongst the latter there are stable periodic orbits, both symmetric and asymmetric with respect to the origin, as well as chaotic attractors. The mosaic structure of the attraction basins is observed. In particular, it is shown that even for neighboring initial conditions can result in distinctly different nonstationary responses asymptotically approach quite different types of attractors. A number of closely neighboring periodic attractors are observed, separated by Hopf bifurcations. Periodic attractors also are observed under special initial conditions in the domains where chaotic behavior is usually expected.

In this paper, the dynamic behavior of the van der Pol-Rayleigh system is studied by using the fast–slow analysis method and the transformation phase portrait method. Firstly, the stability and bifurcation behavior of the equilibrium point of the system are analyzed. We find that the system has no fold bifurcation, but has Hopf bifurcation. By calculating the first Lyapunov coefficient, the bifurcation direction and stability of the Hopf bifurcation are obtained. Moreover, the bifurcation diagram of the system with respect to the external excitation is drawn. Then, the fast subsystem is simulated numerically and analyzed with or without external excitation. Finally, the vibration behavior and its generation mechanism of the system in different modes are analyzed. The vibration mode of the system is affected by both the fast and slow varying processes. The mechanisms of different modes of vibration of the system are revealed by the transformation phase portrait method, because the system trajectory will encounter different types of attractors in the fast subsystem.

A fascinating phenomenon is the self-organization of coupled systems to a whole. This phenomenon is studied for a particular class of coupled oscillatory systems exhibiting so-called simultaneously diagonalizable matrices. For three exemplary systems, namely, an electric circuit, a coupled system of oscillatory neurons, and a system of coupled oscillatory gene regulatory pathways, eigenvectors and amplitude equations are derived. It is shown that for all three systems, only the unstable eigenvectors and their amplitudes matter for the dynamics of the systems on their respective limit cycle attractors. A general class of coupled second-order dynamical oscillators is presented in which stable limit cycles emerging via Hopf bifurcations are solely specified by appropriately defined unstable eigenvectors and their amplitudes. While the eigenvectors determine the orientation of limit cycles in state spaces, the amplitudes determine the evolution of states along those limit cycles. In doing so, it is shown that the unstable eigenvectors define reduced amplitude spaces in which the relevant long-term dynamics of the systems under consideration takes place. Several generalizations are discussed. First, if stable and unstable system parts exhibit a slow-fast dynamics, the fast variables may be eliminated and approximative descriptions of the emerging limit cycle dynamics in reduced amplitude spaces may be again obtained. Second, the principle of reduced amplitude spaces holds not only for coupled second-order oscillators, but can be applied to coupled third-order and higher order oscillators. Third, the possibility to apply the approach to multifrequency limit cycle attractors and other types of attractors is discussed.Graphic abstract

Background: The high nonlinearity of the dopamine circadian rhythms model is seen in the presence of limited cycles that disrupt the circadian rhythms. Limit Cycles originate from Hopf bifurcation points. Bifurcation analysis and Multiobjective nonlinear model predictive control are performed on the dopamine circadian rhythms model. Methods: The MATLAB software MATCONT was used to perform the bifurcation analysis. The Multi-objective Nonlinear Model Predictive Control was performed using the optimization language PYOMO. Results: The Bifurcation analysis reveals Hopf Bifurcation points that produce limit cycles. To eliminate the rhythm disturbing limit cycles the bifurcation parameter is multiplied by an activation factor involving the tanh function. The nonlinearity of the dopamine circadian rhythms model also causes spikes in the control profiles when multiobjective nonlinear model predictive control calculations are performed. The spikes are also eliminated when the control variable is multiplied by the same activation factor. Conclusion: The dopamine circadian rhythms model is shown to have two Hopf bifurcations, which cause limit cycles that can disrupt the circadian rhythms. An activation factor involving the tanh function eliminates the limit cycle causing Hopf bifurcations. This activation factor also removes the spikes that occur in the control profile.