Relaxation oscillations and canard explosion in a predator–prey system of Holling and Leslie types

We give a geometric analysis of relaxation oscillations and canard cycles in a singularly perturbed predator–prey system of Holling and Leslie types. We discuss how the canard cycles are found near the Hopf bifurcation points. The transition from small Hopf-type cycles to large relaxation cycles is also discussed. Moreover, we outline one possibility for the global dynamics. Numerical simulations are also carried out to verify the theoretical results.     

New Penta(tetrathiafulvalenyl)[60]fullerenes for Supramolecular Materials

New penta(organo)fullerenes donor–acceptor systems bearing five tetrathiafulvalene recognition units have been synthesized to promote self‐assemblies similar in appearance to shuttlecocks nested into each other thanks to the conical host cavity created around the fullerene together with the π–π and electronic interactions.     

Unified symmetry of mechano-electrical systems with nonholonomic constraints

The unified symmetry of mechano-electrical systems with nonholonomic constraints are studied in this paper, the definition and the criterion of unified symmetry of mechano-electrical systems with nonholonomic constraints are derived from the Lagrange-Maxwell equations. The Noether conserved quantity, Hojman conserved quantity and Mei conserved quantity are then deduced from the unified symmetry. An example is given to illustrate the application of the results.     

Mei symmetry and Mei conserved quantity of nonholonomic systems with unilateral Chetaev type in Nielsen style

This paper studies the Mei symmetry and Mei conserved quantity for nonholonomic systems of unilateral Chetaev type in Nielsen style. The differential equations of motion of the system above are established. The definition and the criteria of Mei symmetry, loosely Mei symmetry, strictly Mei symmetry for the system are given in this paper. The existence condition and the expression of Mei conserved quantity are deduced directly by using Mei symmetry. An example is given to illustrate the application of the results.     

The multicomponent （2＋1）-dimensional Glachette-Johnson （GJ） equation hierarchy and its super-integrable coupling system

This paper presents a set of multicomponent matrix Lie algebra, which is used to construct a new loop algebra A^-M. By using the Tu scheme, a Liouville integrable multicomponent equation hierarchy is generated, which possesses the Hamiltonian structure. As its reduction cases, the multicomponent （2＋1）-dimensional Glachette-Johnson （G J） hierarchy is given. Finally, the super-integrable coupling system of multicomponent （2＋1）-dimensional GJ hierarchy is established through enlarging the spectral problem.     

A field method for integrating the equations of motion of mechanico--electrical coupling dynamical systems

A field method for integrating the equations of motion for mechanico-electrical coupling dynamical systems is studied. Two examples in mechanico-electrical engineering are given to illustrate this method.     

Perturbation to Mei symmetry and Mei adiabatic invariants for mechanical systems in phase space

For a perturbed mechanical system in phase space, considering d/dt in the structure equation and process of proof including infinitesimal parameter ε obviously, this paper studies the perturbation to Mei symmetry and adiabatic invariants. Firstly, the exact invariant induced directly from the Mei symmetry of the system without perturbation is given. Secondly, based on the concept of high-order adiabatic invariant, the determining equations of the perturbation to Mei symmetry are established, the condition of existence of the Mei adiabatic invariant led by the perturbation to Mei symmetry is obtained, and its form is presented. Lastly, an example is given to illustrate the application of the results.     

Equilibrium points and bifurcation control of a chaotic system

Based on the Routh-Hurwitz criterion, this paper investigates the stability of a new chaotic system. State feedback controllers are designed to control the chaotic system to the unsteady equilibrium points and limit cycle. Theoretical analyses give the range of value of control parameters to stabilize the unsteady equilibrium points of the chaotic system and its critical parameter for generating Hopf bifurcation. Certain nP periodic orbits can be stabilized by parameter adjustment. Numerical simulations indicate that the method can effectively guide the system trajectories to unsteady equilibrium points and periodic orbits.     

Stochastic resonance in an asymmetric bistable system driven by coloured correlated multiplicative and additive noise

This paper investigates the stochastic resonance （SR） phenomenon in an asymmetric system with coupling between multiplicative and additive noise when the coupling between two noise terms is coloured. The approximate expression of signal-to-noise ratio has been obtained by applying the two-state theory and SR exhibits in the bistable system. Moreover, the potential asymmetry r and cross-correlation strength λ can weaken the SR phenomenon, while the cross-correlation time r can strengthen the SR phenomenon.     

Controlling chaos based on an adaptive nonlinear compensator mechanism

The control problems of chaotic systems are investigated in the presence of parametric uncertainty and persistent external disturbances based on nonlinear control theory. By using a designed nonlinear compensator mechanism, the system deterministic nonlinearity, parametric uncertainty and disturbance effect can be compensated effectively. The renowned chaotic Lorenz system subjected to parametric variations and external disturbances is studied as an illustrative example. From the Lyapunov stability theory, sufficient conditions for choosing control parameters to guarantee chaos control are derived. Several experiments are carried out, including parameter change experiments, set-point change experiments and disturbance experiments. Simulation results indicate that the chaotic motion can be regulated not only to steady states but also to any desired periodic orbits with great immunity to parametric variations and external disturbances.