Transient Chaos in Coupled Oscillators with Shape Deformable Potential

The bifurcation analysis of a predator–prey system of Holling and Leslie type with constant-yield prey harvesting is carried out in this paper. It is shown that the model has a Bogdanov–Takens singularity (cusp case) of codimension at least 4 for some parameter values. Various kinds of bifurcations, such as saddle-node bifurcation, Hopf bifurcation, repelling and attracting Bogdanov–Takens bifurcations of codimensions 2 and 3, are also shown in the model as parameters vary. Hence, there are different parameter values for which the model has a limit cycle, a homoclinic loop, two limit cycles, or a limit cycle coexisting with a homoclinic loop. These results present far richer dynamics compared to the model with no harvesting. Numerical simulations, including the repelling and attracting Bogdanov–Takens bifurcation diagrams and corresponding phase portraits, and the existence of two limit cycles or an unstable limit cycle enclosing a stable multiple focus with multiplicity one, are also given to support the theoretical analysis.

This paper investigates the nonlinear dynamics of a flexible tyre coupling via computer modelling and simulation. The research mainly focused on identifying basins of attraction of coexisting solutions of the formulated phenomenological coupling model. On the basis of the derived mathematical model, and by assuming ranges of variability of the control parameters, the areas in which chaotic clutch movement takes place are determined. To identify multiple solutions, a new diagram of solutions (DS) was used, illustrating the number of coexisting solutions and their periodicity. The DS diagram was drawn based on the fixed points of the Poincaré section. To verify the proposed method of identifying periodic solutions, the graphic image of the DS was compared to the three-dimensional distribution of the largest Lyapunov exponent and the bifurcation diagram. For selected values of the control parameter ω, coexisting periodic solutions were identified, and basins of attraction were plotted. Basins of attraction were determined in relation to examples of coexistence of periodic solutions and transient chaos. Areas of initial conditions that correspond to the phenomenon of unstable chaos are mixed with the conditions of a stable periodic solution, to which the transient chaos is attracted. In the graphic images of the basins of attraction, the areas corresponding to the transient and periodic chaos are blurred.

Profits, Cycles and Chaos

Limit cycle bifurcations near homoclinic and heteroclinic loops via stability-changing of a homoclinic loop

Multiple bifurcations in a predator–prey system of modified Holling and Leslie type with double Allee effect and nonlinear harvesting

The motion of a freely rotating prolate spheroid in a simple shear flow of a dilute polymeric solution is examined in the limit of large particle aspect ratio, $\kappa$ . A regular perturbation expansion in the polymer concentration, $c$ , a generalized reciprocal theorem, and slender body theory to represent the velocity field of a Newtonian fluid around the spheroid are used to obtain the $O(c)$ correction to the particle's orientational dynamics. The resulting dynamical system predicts a range of orientational behaviours qualitatively dependent upon $c\, De$ ( $De$ is the imposed shear rate times the polymer relaxation time) and $\kappa$ and quantitatively on $c$ . At a small but finite $c\, De$ , the particle spirals towards a limit cycle near the vorticity axis for all initial conditions. Upon increasing $\kappa$ , the limit cycle becomes smaller. Thus, ultimately the particle undergoes a periodic motion around and at a small angle from the vorticity axis. At moderate $c\, De$ , a particle starting near the flow–gradient plane departs it monotonically instead of spirally, as this plane (a limit cycle at smaller $c\, De$ ) obtains a saddle and an unstable node. The former is close to the flow direction. Upon further increasing $c\, De$ , the saddle node changes to a stable node. Therefore, depending upon the initial condition, a particle may either approach a periodic orbit near the vorticity axis or obtain a stable orientation near the flow direction. Upon further increasing $c\, De$ , the limit cycle near the vorticity axis vanishes, and the particle aligns with the flow direction for all starting orientations.

The emergence of multistability in a simple three-dimensional autonomous oscillator is investigated using numerical simulations, calculations of Lyapunov exponents and bifurcation analysis over a broad area of two-dimensional plane of control parameters. Using Neimark–Sacker bifurcation of 1:1 limit cycle as the starting regime, many parameter islands with the coexisting attractors were detected in the phase diagram, including the coexistence of torus, resonant limit cycles and chaos; and transitions between the regimes were considered in detail. The overlapping between resonant limit cycles of different winding numbers, torus and chaos forms the multistability.

Biological nervous system function is closely related to its dynamical behaviors, and some dynamical phenomena observed in biological systems can be detected in the simplified neural models. In this paper, the chaotic dynamics in a three-neuron-based Hopfield neural network (HNN) with stimulation of electromagnetic radiation is investigated. The neural network is modeled by utilizing a flux-controlled memristor to describe the effects of electromagnetic field on neurons. The simple neural model affected by electromagnetic radiation does not contain any equilibrium points, but can induce coexisting infinitely many hidden attractors, such as hyperchaos, transient hyperchaos, period, quasi-period, chaos as well as transient chaos with different chaotic times. In particular, the dynamics of hidden extreme multistability with hyperchaos and transient chaos in the neural network highly depends on the system parameters and state initial values. The coexistence of multiple hidden attractors is revealed via applying a host of numerical analysis methods including phase plots, time sequence waveforms, bifurcation diagrams, Lyapunov exponents and attraction basins. Besides, a HNN-based circuit consisting of commercially available electronic elements is designed to verify the theoretical analysis. Hardware measurement and MULTISIM simulation results are basically consistent with MATLAB numerical simulation results.

In this paper we investigate the dynamical behaviour of a cubic Liénard system with global parameters. After analysing the qualitative properties of all the equilibria and judging the existences of limit cycles and homoclinic loops for the whole parameter plane, we give the bifurcation diagram and phase portraits. Phase portraits are global if there exist limit cycles and local otherwise. We prove that parameters lie in a connected region, not just on a curve, usually in the parameter plane when the system has one homoclinic loop. Moreover, for global parameters we give a positive answer to conjecture 3.2 of (1998 Nonlinearity 11 1505–19) in the case of exactly two equilibria about the existence of some function whose graph is exactly the surface of double limit cycles.

Example of a quadratic system with two cycles appearing in a homoclinic loop bifurcation

Homoclinic bifurcation is a difficult and important topic of bifurcation theory. As we know, a general theory for a homoclinic loop passing through a hyperbolic saddle was established by [Roussarie, 1986]. Then the method of stability-changing to find limit cycles near a double homoclinic loop passing through a hyperbolic saddle was given in [Han & Chen, 2000], and further developed by [Han et al., 2003; Han & Zhu, 2007]. For a homoclinic loop passing through a nilpotent saddle there are essentially two different cases, which we distinguish by cuspidal type and smooth type, respectively. For the cuspidal type a general theory was recently established in [Zang et al., 2008]. In this paper, we consider limit cycle bifurcation near a double homoclinic loop passing through a nilpotent saddle by studying the analytical property of the first order Melnikov functions for general near-Hamiltonian systems and obtain the conditions for the perturbed system to have 8, 10 or 12 limit cycles in a neighborhood of the loop with seven different distributions. In particular, for the homoclinic loop of smooth type, a general theory is obtained as a consequence. We finally consider some polynomial systems and find a lower bound of the maximal number of limit cycles as an application of our main results.

It was shown in [Li & Xiao, 2007] that in a predator–prey model of Leslie type with simplified Holling type IV functional response some complex bifurcations can occur simultaneously for some values of parameters, such as codimension 1 subcritical Hopf bifurcation and codimension 2 Bogdanov–Takens bifurcation. In this paper, we show that for the same model there exists a unique degenerate positive equilibrium which is a degenerate Bogdanov–Takens singularity (focus case) of codimension 3 for other values of parameters. We prove that the model exhibits degenerate focus type Bogdanov–Takens bifurcation of codimension 3 around the unique degenerate positive equilibrium. Numerical simulations, including the coexistence of three hyperbolic positive equilibria, two limit cycles, bistability states (one stable equilibrium and one stable limit cycle, or two stable equilibria), tristability states (two stable equilibria and one stable limit cycle), a stable limit cycle enclosing a homoclinic loop, a homoclinic loop enclosing an unstable limit cycle, or a stable limit cycle enclosing three unstable hyperbolic positive equilibria for various parameter values, confirm the theoretical results.

Bifurcations of limit cycles in equivariant quintic planar vector fields

This study explores a novel two-dimensional discrete-time ratio-dependent Holling-Tanner predator-prey model, incorporating the impact of the Fear effect on the prey population. The study focuses on identifying stationary points and analyzing bifurcations around the positive fixed point, with an emphasis on their biological significance. Our examination of bifurcations at the interior fixed point uncovers a variety of generic bifurcations, including one-parameter bifurcations, period-doubling, and Neimark-Sacker bifurcations. To further understand NS bifurcation, we establish non-degeneracy condition. The system's bifurcating and fluctuating behavior is managed using Ott-Grebogi-Yorke (OGY) control technique. From an ecological perspective, these findings underscore the substantial role of the Fear effect in shaping predator-prey dynamics. The research is extended to a networked context, where interconnected prey-predator populations demonstrate the influence of coupling strength and network structure on the system's dynamics. The theoretical results are validated through numerical simulations, which encompass local dynamical classifications, calculations of maximum Lyapunov exponents, phase portrait analyses, and bifurcation diagrams.

We theoretically investigate a two-component reaction-diffusion system that exhibits transient spatio-temporal chaos before reaching a periodic attractor. The transient times grow exponentially with the size of the system. The chaotic behaviour is characterized by the calculation of local Lyapunov exponents and the propagation velocity of a localized perturbation. Both the maximum positive Lyapunov exponent and the propagation velocity appear to be independent of the system size.