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Abstract A prey–predator model with constant-effort harvesting on the prey and predators is investigated in this paper. First, we discuss the number and type of the equilibria by analyzing the equations of equilibria and the distribution of eigenvalues. Second, with the rescaled harvesting efforts as bifurcation parameters, a subcritical Hopf bifurcation is exhibited near the multiple focus and a Bogdanov–Takens bifurcation is also displayed near the BT singularity by analyzing the versal unfolding of the model. With the variation of bifurcation parameters, the system shows multi-stable structure, and the attractive domains for different attractors are constituted by the stable and unstable manifolds of saddles and the limit cycles bifurcated from Hopf and Bogdanov–Takens bifurcations. Finally, a cusp point and two generalized Hopf points are found on the saddle-node bifurcation curve and the Hopf bifurcation curves, respectively. Several phase diagrams for parameters near one of the generalized Hopf points are exhibited through the generalized Hopf bifurcation.

A prey–predator model with constant-effort harvesting on the prey and predators is investigated in this paper. First, we discuss the number and type of the equilibria by analyzing the equations of equilibria and the distribution of eigenvalues. Second, with the rescaled harvesting efforts as bifurcation parameters, a subcritical Hopf bifurcation is exhibited near the multiple focus and a Bogdanov–Takens bifurcation is also displayed near the BT singularity by analyzing the versal unfolding of the model. With the variation of bifurcation parameters, the system shows multi-stable structure, and the attractive domains for different attractors are constituted by the stable and unstable manifolds of saddles and the limit cycles bifurcated from Hopf and Bogdanov–Takens bifurcations. Finally, a cusp point and two generalized Hopf points are found on the saddle-node bifurcation curve and the Hopf bifurcation curves, respectively. Several phase diagrams for parameters near one of the generalized Hopf points are exhibited through the generalized Hopf bifurcation. [ABSTRACT FROM AUTHOR]

In this note, some sufficient conditions on the ergodic sensitivity of dynamical systems are obtained, improving the main results in [Q.-L. Huang, Y.-M. Shi, L.-J. Zhang, Appl. Math. Lett., 39 (2015), 31-34] and [R.-S. Li, Y.-M. Shi, Nonlinear Anal., 72 (2010), 2716-2720]. Moreover, it is proved that under these conditions, the second Lyapunov number of a dynamical system is equal to the diameter of its state space. [ABSTRACT FROM AUTHOR]

Abstract: The phase portraits, existence and uniqueness of stable limit cycles and Hopf bifurcations of the well-known Holling–Tanner models for predator–prey interactions are studied. The ranges of the parameters involved are provided under which the unique interior equilibrium can be determined to be a stable (or an unstable) node or focus. The Hopf bifurcations and the existence and uniqueness of stable limit cycles of the models are obtained by computing the Lyapunov number involved. Our results confirm some previous results observed and suggested from the real ecological systems. [Copyright &y& Elsevier]

This article analyzes a mathematical model for some aeroelastic oscillators in physiology, based on a previous representation for the vocal folds at phonation. The model characterizes the oscillation as superficial wave propagating through the tissues in the direction of the flow, and consists of a functional differential equation with advanced and delay arguments. The analysis shows that the oscillation occurs at a Hopf bifurcation, at which the energy absorbed from the flow overcomes the energy dissipated in the tissues. The bifurcation value of the flow pressure increases linearly with the tissue damping and the oscillation frequency. Also, it is minimum when the phase delay of the superficial wave to travel along the tissues is π, and increases indefinitely when the delay tends to 0 and to 2π. [ABSTRACT FROM AUTHOR]

A system of logistic map coupled with slowly driven items or another map is investigated in this paper. Several artificial behaviors emerge if the relation is not kept: α β < γ. Here α is number of computational precision digit, for instance, for double precision which has a unit round-off of 2 × 10−16. And β is parameter characterized of maximum of local dynamical expansion ability β or contraction ability 1/β. γ is attractor size of the system. β can be extracted from concept of the Lyapunov number, which is closely related to the well-known Lyapunov exponent. [ABSTRACT FROM AUTHOR]

In reactive sputtering, the introduction of reactive gas would create a hysteresis transition from metal to compounds (oxides, nitrides and carbides, etc.) in both target and substrate. The hysteresis transition is characterized by a sudden change in partial pressure, cathode voltage, sputtering rate and fraction of compound formation. Therefore, the stability is an important issue of process control. In this article, a simple mathematical model based on Berg''s original proposal is used to study the stability of steady state solutions. In order to facilitate the analysis, several non-dimensional parameters are identified and used for formulation. The stability is then investigated in two folds. For a small variation away from the steady state, the linear perturbation method is utilized to reduce the model to an eigenvalue problem. For large deviation from the steady state, a full numerical analysis is employed to obtain the steady state. Results show that an unsteady system converges to steady states relatively fast at low inflow rates and the positive range of eigenvalue are coherent with the presence of hysteresis loop. With the aid of the eigenvalue, it is also found that when the chemical reaction on the substrate is moderate, a higher sputter yield of the compound leads to a more stable steady state at lower inflow rates. Finally, a convergence parameter similar to the Lyapunov number is introduced for full numerical analysis to give a qualitative measure for the stability at steady state. [Copyright &y& Elsevier]

In this paper, with the theory of nonlinear dynamic systems, It is analyzed that the dynamic behavior and the predictability for the monthly mean variations of the sunspot relative number recorded from January 1891 to December 1996. In the progress, the fractal dimension (D=3.3±0.2) for the variation process was computed. This helped us to determine the embedded dimension [2×D+1]=7. By computing the Lyapunov index (λ1=0.863), it was indicated that the variation process is a chaotic system. The Kolmogorov entropy (K=0.0260) was also computed, which provides, theoretically, the predicable time scale. And at the end, according to the result of the analysis above, an experimental predication is made, whose data was a part cut from the sample data. [ABSTRACT FROM AUTHOR]