International Journal of Bifurcation and Chaos

We present a method that generalizes the periodic orbit dividing surface construction for Hamiltonian systems with three or more degrees of freedom. We construct a torus using as a basis a periodic orbit and we extend this to a ($2n-2$)-dimensional object in the ($2n-1$)-dimensional energy surface. We present our methods using benchmark examples for two and three degrees of freedom Hamiltonian systems to illustrate the corresponding algorithm for this construction. Towards this end, we use the normal form quadratic Hamiltonian system with two and three degrees of freedom. We found that the periodic orbit dividing surface can provide us the same dynamical information as the dividing surface constructed using normally hyperbolic invariant manifolds. This is significant because, in general, computations of normally hyperbolic invariant manifolds are very difficult in Hamiltonian systems with three or more degrees of freedom. However, our method avoids this computation and the only information that we need is the location of one periodic orbit.

There are few explicit examples in the literature of vector fields exhibiting observable chaos that may be proved analytically. This paper reports numerical experiments performed for an explicit two-parameter family of $𝕊𝕆(2)\oplus {\mathbb{Z}}_2$-symmetric vector fields whose organizing center exhibits an attracting heteroclinic network linking two saddle-foci. Each vector field in the family is the restriction to ${𝕊}^3$ of a polynomial vector field in ${ℝ}^4$. We investigate global bifurcations due to symmetry-breaking and we detect strange attractors via a mechanism called Torus-Breakdown. We explain how an attracting torus gets destroyed by following the changes in the unstable manifold of a saddle-focus.

Although a complete understanding of the corresponding bifurcation diagram and the mechanisms underlying the dynamical changes is out of reach, we uncover complex patterns for the symmetric family under analysis, using a combination of theoretical tools and computer simulations. This article suggests a route to obtain rotational horseshoes and strange attractors; additionally, we make an attempt to elucidate some of the bifurcations involved in an Arnold tongue.

Considering the mature condition of any individual to have eaten a specific amount of food during the entire period that it can spend at its immature stage, we propose a size-structured model by a first-order quasi-linear partial differential equation. The model can be firstly reduced to a single state-dependent delay differential equation and then to a constant delay differential equation. The state-dependent delay represents intra-specific competition among individuals for limited food resources. A complete analysis of the global dynamics on the positivity and boundedness of solutions, global stability for each equilibrium and Hopf bifurcation is carried out. Our results imply that the delay leads to instability that is shown by a simple example of a certain structured population model.

As the only nonlinear component for symmetric cryptography, S-Box plays an important role. An S-Box may be vulnerable because of the existence of fixed point, reverse fixed point or short iteration cycles. To construct a keyed strong S-Box, first, a 2D enhanced quadratic map (EQM) was constructed, and its dynamic behaviors were analyzed through phase diagram, Lyapunov exponent, Kolmogorov entropy, bifurcation diagram and randomness testing. The results demonstrated that the state points of EQM have uniform distribution, ergodicity and better randomness. Then a keyed strong S-Box construction algorithm was designed based on EQM, and the fixed point, reverse fixed point, and short cycles were eliminated. Experimental results verified the algorithm’s feasibility and effectiveness.

The bifurcations and chaos in a system of two coupled sigmoidal neurons with periodic input are revisited. The system has no self-coupling and no inherent limit cycles in contrast to the previous studies and shows simple bifurcations qualitatively different from the previous results. A symmetric periodic solution generated by the periodic input underdoes a pitchfork bifurcation so that a pair of asymmetric periodic solutions is generated. A chaotic attractor is generated through a cascade of period-doubling bifurcations of the asymmetric periodic solutions. However, a symmetric periodic solution repeats saddle-node bifurcations many times and the bifurcations of periodic solutions become complicated as the output gain of neurons is increasing. Then, the analysis of border collision bifurcations is carried out by using a piecewise constant output function of neurons and a rectangular wave as periodic input. The saddle-node, the pitchfork and the period-doubling bifurcations in the coupled sigmoidal neurons are replaced by various kinds of border collision bifurcations in the coupled piecewise constant neurons. Qualitatively the same structure of the bifurcations of periodic solutions in the coupled sigmoidal neurons is derived analytically. Further, it is shown that another period-doubling route to chaos exists when the output function of neurons is asymmetric.

Crack in gears impacts the dynamic response of wind turbine multistage gear system, which also influences the safe operation of wind turbine. A translational–torsional nonlinear dynamic model of the multistage gear system is proposed with root crack fault. The model considers the effects of sun gear support, time-varying mesh stiffness, gear backlash and other factors. The mesh stiffness with root crack is analyzed by using potential energy method. Based on the Runge–Kutta method, the system responses are obtained with multiple parameters changing. The nonlinear dynamic features of the cracked and normal system are compared by bifurcation diagram, time series, phase trajectory, Poincaré map, spectrum diagram and corresponding three-dimensional diagrams. The analyses show the effects of input torque, backlash, crack occurrence and evolution on the system dynamic behaviors, and the effect of crack fault on the gear system response is further verified by experiment. The results provide a theoretical basis for the cognition of fault mechanism and fault diagnosis of wind turbine gearbox.

In this paper, we consider a reduction of the Gierer–Meinhardt Activator–Inhibitor model. In the absence of diffusion, we determine the global dynamics of the homogeneous system. Then, we study the effect of the diffusion constants on the stability of a homogeneous steady state. By choosing a proper bifurcation parameter, we prove that, under some suitable conditions on the parameters, a generalized Hopf bifurcation occurs in the inhomogeneos model. We compute the normal form of this bifurcation up to the fifth order. Furthermore, the direction of the Hopf bifurcation is obtained by the normal form theory. Finally, we provide some numerical simulations to justify our theoretical results.

In ecology, the predator’s impact goes beyond just killing the prey. In the present work, we explore the role of fear in the dynamics of a discrete-time predator-prey model where the predator-prey interaction obeys Holling type-II functional response. Owing to the increasing strength of fear, the system becomes stable from chaotic oscillations via inverse Neimark–Sacker bifurcation. Extensive numerical simulations are carried out to investigate the intricate dynamics for the organization of periodic structures in the bi-parameter space of the system. We observe fear induced multistability between different pairs of coexisting heterogeneous attractors due to the overlapping of multiple periodic domains in the bi-parameter space. The basin sets of the coexisting attractors are obtained and discussed at length. Multistability in the predator-prey system is important because the dynamics of the predator and prey populations in the critical parameter zone becomes uncertain.

Let $(S,\tau )$ be a nontrivial compact metric space with metric $\tau$ and $g:S\to S$ be a continuous self-map, $\mathcal{ℬ}(S)$ be the sigma-algebra of Borel subsets of $S$, and $\nu$ be a Borel probability measure on $(S,\mathcal{ℬ}(S))$ with $\nu (V)\in (0,+\infty )$ for any open subset $V\ne \varnothing$ of $S$. This paper proves the following results :

• (1)If the pair $(g,\nu )$ has the property that for any $\beta >0$, there is $r(\beta )>0$ such that
  $$\nu \left(\left\{p\in S:\left|\frac{1}{m}\sum _{i=0}^{m-1}{{𝒳}}_V(g^i(p))-\int {{𝒳}}_{V}d\nu \right|>\beta \right\}\right)\le e^{-mr(\beta )},$$
  for any open subset $V\ne \varnothing$ of $S$ and all $n\ge 1$ sufficiently large (where ${{𝒳}}_V$ is the characteristic function of the set $V$), then the following hold :
  • (a)The map $g$ is topologically ergodic.
  • (b)The upper density $\overline{\mu }(N_g(p,V))$ of $N_g(p,V)$ is positive for any open subset $V\ne \varnothing$ of $S$, where $N_g(p,V)=\{m\in \{0,1,\dots \}:g^m(p)\in V\}$.
  • (c)There is a $g$-invariant Borel probability measure $\nu$ having full support (i.e. $\mathrm{\text{supp}}(\nu )=S$).
  • (d)Sensitivity of the map $g$ implies positive lower density sensitivity, hence ergodical sensitivity.
• (2)If $\underline{\mu }(N_g(A,B))=1$ for any two nonempty open subsets $A,B$, then there exists $\lambda >0$ satisfying $\underline{\mu }(N_g(C,\lambda ))=1$ for any nonempty open subset $C\subset S$, where $N_g(C,\lambda )=\{l\in \{0,1,\dots \}:$ there exist $a,b\in C$ with $\tau (g^l(a),g^l(b))>\lambda \}$.

Due to high computational cost, traditional encryption algorithms are not suitable for the environments in which resources are limited. In view of the above problem, we first propose a combined chaotic map to increase the chaotic interval and Lyapunov exponent of the existing one-dimensional chaotic maps. Then, an S-box based on the proposed combined chaotic map is constructed. The performances of the designed S-box, such as bijection, nonlinearity, strict avalanche criteria, differential uniformity, the bits independence criterion, and the linear approximation probability, are tested to show that it has better cryptographic performances. Finally, we present a lightweight block encryption algorithm by using the above S-box. The algorithm is based on the generalized Feistel structure and SPN structure. In addtion, the processes of encryption and decryption of our algorithm are almost the same, which reduces the complexity of algorithm implementation. The experimental results show that the proposed encryption algorithm meets the requirements of lightweight algorithms and has good cryptographic characteristics.

The temperature change is a non-negligible factor in examining the vibration behaviors of the cable structures. For this reason, the paper aims at investigating the thermal effects on suspended cables’ resonant responses considering two-to-one internal resonances. Firstly, a nonlinear continuous condensed model of the suspended cable under periodic excitation in thermal environments is adopted. Then, a multidimensional discretized model is constructed via the Galerkin method. Following the multiple scaling procedure, the modulation equations with both polar and Cartesian forms are obtained, which are solved numerically. A complete dynamic scenario is presented through bifurcation diagrams, phase portraits, time history curves, Fourier spectra, and Poincaré sections in three internal resonant cases. Numerical examples show that a small change in the static configuration due to thermal effects induces some noticeable changes in dynamic behaviors. The response amplitude, the nonlinear spring behavior, the resonant and stability region, the multi-periodic and chaotic motions are all dependent on temperature changes. Additional Hopf bifurcations might be found due to temperature changes, and it may lead to some more complicated dynamic characteristics. A good agreement between the perturbation and numerical solutions is observed to confirm the results’ correctness and accuracy.

For a family of two-dimensional predator–prey models of Gause type, we investigate the simultaneous occurrence of a center singularity and a limit cycle. The family is characterized by the fact that the functional response is nonanalytical and exhibits group defense. We prove the existence and uniqueness of the limit cycle using a new theorem for Liénard systems. The new theorem gives conditions for the uniqueness of a limit cycle which surrounds a period annulus. The results of this paper provide a mechanism for studying the global behavior of solutions to Gause systems through bifurcation of an integrable system which contains a center and a limit cycle.

Aiming to clarify the leading roles of new-type agricultural business entities (abbreviated as NABEs) to small-scale farmers, integrated game models between NABEs and small-scale farmers are designed to verify the impact of scale economy and investment spillover on the equilibrium points of the game system. The influence of the investment spillover rate and the decision-making adjustment speed on the stability of the system are emphatically discussed. Numerical simulation shows that, with the increase of small-scale farmers’ decision-making adjustment speed, the system would successively show the phenomena of stability, period doubling bifurcation, chaos and discreteness. In the Cournot game, the two sides’ decision-making results such as investment intensity, selling price and eventual profits vary in the opposite direction. In the Stackelberg game of the basic mode, the two sides’ decision-making results are not evidently changed, and NABEs’ investment intensity, selling price and eventual profits are higher than those of small-scale farmers. In the order mode, system improvement can be realized by controlling the investment spillover rate. The research results indicate that with the increase of the adjustment speed of small-scale farmers’ decision-making, the repeated game decision-making aggravates the instability of the Cournot game system. This paper finds that the order pattern can make up for the scale weakness of small-scale farmers, and finally achieve a win-win situation for decision makers in the case of uncertain demands.

This paper focuses on the leader selection from the leader-based collective bargaining system, where buyers form an alliance and designate one of them as the leader to bargain with the supplier for a lower wholesale price of their common component. We construct the dynamic bargaining system consisting of two heterogeneous buyers and one supplier to analyze the influence of the enterprise nature and bargaining power on the leader selection. It was proved that the buyer with stronger bargaining power should be the leader. However, we find that, when the buyers are heterogeneous, the result may be different. In order to explore which factor plays a more important role in the leader selection, we design two rounds of bargaining for comparison. The interesting results imply that whether the bargaining power will reverse the leader selection in the first round depends on its growth rate. The nonlinear dynamics theory is also introduced to analyze the complex behaviors in the dynamic bargaining system. We analyze the influence of adjustment speed on the dynamic bargaining system and obtain the conditions required to maintain system stability. Considering the significance of system stability, the delayed feedback control mechanism is adopted to drive chaos back to stability.

In this paper, we formulate a new-age structured heroin transmission model with respect to the age of vaccination which structures the vaccine wanes rate $\alpha (a)$ and infection ratio of vaccination individuals $\sigma (a)$. The well-posedness and the basic reproduction number ${{ℛ}}_0$ of our model are first presented. If ${{ℛ}}_0<1$, the drug-free steady state ${{ℰ}}^0$ is locally stable and there will be multiple positive steady states due to the imperfect vaccine. If ${{ℛ}}_0>1$, there is a unique drug spread steady state, and our model is uniformly persistent. To reveal the dynamics of our model in detail, we carry out a further analysis in some special cases, including the backward and forward bifurcation results of our model when $\alpha (a)=\alpha$ and $\sigma (a)=\sigma$, and the unique drug-spread steady state’s stability when ${{ℛ}}_0>1$. Finally, a brief conclusion and discussion are presented.

In this paper, we establish a predator–prey model with focus on the Allee effect and prey group defense. The positivity and boundedness of the model, existence of equilibrium point, and stability change caused by Allee effect are studied. Bifurcation (transcritical bifurcation, Hopf bifurcation) analysis is discussed, and the direction of Hopf bifurcation is determined by calculating the first Lyapunov number. Then we introduce delay into the original model and consider the influence of delay on the stability of the model. By selecting delay as the bifurcation parameter, we obtain the existence conditions of Hopf bifurcation and the direction of Hopf bifurcation. Finally, we verify the theoretical analysis by numerical simulation. Considering both the Allee effect and the prey group defense, the dynamic behavior near the origin becomes more complex than only considering Allee effect or prey group defense in the model. Allee effect can bring the risk of extinction and the change of stability, and the delay effect can make the stable coexistence equilibrium unstable and lead to periodic oscillation.

We study bifurcations of limit cycles arising after perturbations of a special piecewise smooth system, which has a center and a homoclinic loop. By using the Picard–Fuchs equation, we give an upper bound of the maximum number of limit cycles bifurcated from the period annulus between the center and the homoclinic loop. Furthermore, by applying the method of first-order Melnikov function we obtain a lower bound of the maximum number of limit cycles bifurcated from the center.

Generic results for degenerate Chenciner (generalized Neimark–Sacker) bifurcation are obtained in the present work. The bifurcation arises from two-dimensional discrete-time systems with two independent parameters. We define in this work a new transformation of parameters, which enables the study of the bifurcation when degeneracy occurs. By the four bifurcation diagrams we obtained, new behaviors hidden by the degeneracy are brought to light.