Narrow Results

Based on the Riemann theta function, the Hirota bilinear method is extended to directly construct a kind of explicit quasi-periodic solution of the Toda lattice equation. The asymptotic property of the quasi-periodic solution is analyzed in detail. It is of interest to see that the well-known soliton solution can be obtained as a limit of the quasi-periodic solution.

This paper investigates the bifurcations due to time delay in the feedback control system with excitation. Based on an self-sustained oscillator, the delayed velocity feedback control system is proposed. For the case without excitation, the stability of the trivial equilibrium is discussed and the condition under which the equilibrium loses its stability is obtained. This leads to a critical stability boundary where Hopf bifurcation or periodic solutions may occur. For the case with excitation, the main attention is focused on the effect of time delay on the obtained periodic solution when primary resonance occurs in the system under consideration. To this end, the control system is changed to be a functional differential equation. Functional analysis is carried out to obtain the center manifold and then a perturbation approach is used to find periodic solutions in a closed form. Moreover, the unstable regions for the limit cycles are also obtained, predicting the occurrence of some complex behaviors. Numerical simulations are employed to find the routes leading to quasi-periodic motions as the time delay is varied. It has been found that: (i) Time delay can be used to control bifurcations; and (ii) time delay can be applied to generate bifurcations. This indicates that time delay may be used as a "switch" to control or create complexity for different applications.

This paper studies the nonlinear equation for ventilation flows in a single opening enclosure. Periodic solutions, the global attractive region, periods and amplitudes of the periodic solutions, and quasi-periodic solutions in the nonlinear equation are investigated.

By using the continuation theorem of coincidence degree theory and constructing suitable Lyapunov functions, we study the existence, uniqueness and global exponential stability of periodic solution for continuous-time neural networks under pure structural variations with impulsive perturbations:

For differential equation in the Chazy class IX, their corresponding three-dimensional system is studied in this paper by using dynamical system methods and Cosgrove's results. In a level set, the exact explicit parametric representations of a heteroclinic cycle family and uncountably infinitely many periodic solutions as well as quasi-periodic solutions, are all obtained.

In this study, we investigate the bifurcation of the quasi-periodic solution in a ring of three-coupled, bistable oscillators. The quasi-periodic solution under study is a propagating wave solution with three-phase synchronized envelope, and we name it "ICC3ϕ" which means invariant closed curve for which time waveforms have three-phase synchronized envelope. We obtain the two-parameter bifurcation diagram of ICC3ϕ with respect to the coupling factor α versus the nonlinear strength ϵ. Consequently, we detect several Arnold tongues showing various periodic solutions of ICC3ϕ. By investigating the bifurcation on the boundary of these Arnold tongues, we clarify the transition from periodic to quasi-periodic solution of ICC3ϕ or the inverse. Specifically, the transition of the period-4, period-5, and period-7 solutions to ICC3ϕ is due to a saddle-node bifurcation, which has no hysteresis. In contrast, the transition from the period-3 solution to ICC3ϕ is a combination of subcritical pitchfork and heteroclinic bifurcations; hence, it has hysteresis. We draw diagrams showing the connection of unstable manifolds of heteroclinic bifurcation.

In our preceding study, we investigated the appearance and disappearance scenario of (quasi-periodic) switching solution in a ring of six-coupled, bistable oscillators. This system was expressed as twelve-dimensional phase space, which was divided into linear subspace H and its complimentary linear subspace H^⊥. We investigated the stability of switching solution in H by calculating the conditional Lyapunov exponent to the direction of H^⊥. As a result, we found a critical coupling factor α_crt at which switching solution lost its stability in the direction of H^⊥. However, we did not know what type of bifurcation occurred at this point. In this study, we elucidate that switching solution causes a supercritical pitchfork bifurcation at α_crt.

In continuous-time dynamical systems, a periodic orbit becomes a fixed point on a certain Poincaré section. The eigenvalues of the Jacobian matrix at this fixed point determine the local stability of the periodic orbit. Analogously, a quasi-periodic orbit (2-torus) becomes an invariant closed curve (ICC) on a Poincaré section. From the Lyapunov exponents of an ICC, we can determine the time average of the exponential divergence rate of the orbit, which corresponds to the eigenvalues of a fixed point. We denote the Lyapunov exponent with the smallest nonzero absolute value as the Dominant Lyapunov Exponent (DLE). A local bifurcation manifests as a crossing or touch of the DLE locus with zero. However, the type of bifurcation cannot be determined from the DLE. To overcome this problem, we define the Dominant Lyapunov Bundle (DLB), which corresponds to the dominant eigenvectors of a fixed point. We prove that the DLB of a 1-torus in a map can be classified into four types: A⁺ (annulus and orientation preserving), A^- (annulus and orientation reversing), M (Möbius band), and F (focus). The DLB of a 2-torus in a flow can be classified into three types: A⁺ × A⁺, A^- × M (equivalently M × A^- and M × M), and F × F. From the results, we conjecture the possible local bifurcations in both cases. For the 1-torus in a map, we conjecture that type A⁺ and A^- DLBs correspond to a saddle-node and period-doubling bifurcations, respectively, whereas a type M DLB denotes a double-covering bifurcation, and type F relates to a Neimark–Sacker bifurcation. Similarly, for the 2-torus in a flow, we conjecture that type A⁺ × A⁺ DLBs correspond to saddle-node bifurcations, type A^- × M DLBs to double-covering bifurcations, and type F × F DLBs to the Neimark–Sacker bifurcations. After introducing the mathematical concepts, we provide a DLB-calculating algorithm and illustrate all of the above bifurcations by examples.

In this paper, we investigate the codimension-two double Hopf bifurcation in delay-coupled van der Pol–Duffing oscillators. By using normal form theory of delay differential equations, the normal form associated with the codimension-two double Hopf bifurcation is calculated. Choosing appropriate values of the coupling strength and the delay can result in nonresonance and weak resonance double Hopf bifurcations. The dynamical classification near these bifurcation points can be explicitly determined by the corresponding normal form. Periodic, quasi-periodic solutions and torus are found near the bifurcation point. The numerical simulations are employed to support the theoretical results.

In this paper, we consider the exact explicit solutions for the famous generalized Hénon–Heiles (H–H) system. Corresponding to the three integrable cases, on the basis of the investigation of the dynamical behavior and level curves of the planar dynamical systems, we find all possible explicit exact parametric representations of solutions in the invariant manifolds of equilibrium points in the four-dimensional phase space. These solutions contain quasi-periodic solutions, homoclinic solutions, periodic solutions as well as blow-up solutions. Therefore, we answer the question: what are the flows in the center manifolds and homoclinic manifolds of the generalized Hénon–Heiles (H–H) system. As an application of the above results, we consider the traveling wave solutions for the coupled $(n+1)$-dimensional Klein–Gordon–Schrödinger Equations with quadratic power nonlinearity.

This paper considers an SIRS epidemic model with proportional impulsive vaccination, which may inherently oscillate. We study the ratio-dependent impulsive control and obtain the conditions about the basic reproductive number for which the epidemic-elimination solution is globally asymptotic. On the other hand, if the epidemic turns out to be endemic, we study numerically the influences of impulsive vaccination on the periodic oscillation of a system without impulsion and find sophisticated phenomena such as chaos in this case.

We study an impulsive delay differential predator–prey model with Holling type II functional response. The stability of the trivial equilibrium is analyzed by means of impulsive Floquet theory providing a sufficient condition for extinction. Using coincidence degree theory we show the existence of positive periodic solutions. The system is then analyzed numerically, revealing that the presence of delays and impulses may lead to chaotic solutions, quasi-periodic solutions, or multiple periodic solutions. Several simulations and examples are presented.

In this paper, the backward and forward Neumann type systems are generalized to deduce the quasi-periodic solutions for a negative-order integrable system of 2-component KdV equation. The 2-component negative-order KdV (2-nKdV) equation is depicted as the zero-curvature representation of two spectral problems. It follows from a symmetric constraint that the 2-nKdV equation is reduced to a pair of backward and forward Neumann type systems, where the involutive solutions of Neumann type systems yield the finite parametric solutions of 2-nKdV equation. The negative-order Novikov equation is given to specify a finite-dimensional invariant subspace for the 2-nKdV flow. With a spectral curve given by the Lax matrix, the 2-nKdV flow is linearized on the Jacobi variety of a Riemann surface, which leads to the quasi-periodic solutions of 2-nKdV equation by using the Riemann-Jacobi inversion.

In this paper the present author gives out his some works on nonlinear wave equation using KAM theory. In Section 1, some backgrounds are given out for solving the nonlinear wave (NLW) equation via KAM theory. In Section 2, Theorem 2.1 shows that there are many invariant tori for the NLW equation with any non-vanishing prescribed potential. In Section 3, Theorems 3.1 and 3.2 show that there are many invariant tori of any dimension for the completely resonant NLW equation. This answers an open problem proposed by several authors.