Stable Quasi-periodic Solutions Research Articles

Double Hopf bifurcation analysis of time delay coupled active control system

In this paper, the stability analysis of double Hopf bifurcation is carried out based on the time delay coupled active control system by using the related theory of bifurcation analysis. We got the stability switching region of the system with respect to time delay. The time delay and coupling strength are selected as the bifurcation parameters, then the normal form of the coupled time delay active control system is obtained by using the multi-scale method, and the dynamic behavior near the bifurcation point is obtained. So the parameter region of the stable periodic solution and the stable quasi-periodic solution are obtained. The analytical solution of the system is then solved by using the multi-frequency homotopy analysis method (MFHAM) to verify the existence of the unstable periodic solution. Finally, numerical simulation is executed to verify the theoretical results.

Invariant Tori for a Two-Dimensional Completely Resonant Beam Equation with a Quintic Nonlinear Term

This paper focuses on a two-dimensional completely resonant beam equation with a quintic nonlinear term. This means studying u t t + Δ 2 u + ε f u = 0 , x ∈ T 2 , t ∈ ℝ , under periodic boundary conditions, where ε is a small positive parameter and f u is a real analytic odd function of the form f u = f 5 u 5 + ∑ i ^ ≥ 3 f 2 i ^ + 1 u 2 i ∧ + 1 , f 5 ≠ 0 . It is proved that the equation admits small-amplitude, Whitney smooth, linearly stable quasiperiodic solutions on the phase-flow invariant subspace ℤ † 2 = r = r 1 , r 2 , r 1 ∈ 4 ℤ − 1 , r 2 ∈ 4 ℤ . Firstly, the corresponding Hamiltonian system of the equation is transformed into an angle-dependent block-diagonal normal form by using symplectic transformation, which can be achieved by selecting the appropriate tangential position. Finally, the existence of a class of invariant tori is proved, which implies the existence of quasiperiodic solutions for most values of frequency vector by an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional Hamiltonian systems.

Complex dynamics in the improved Koren–Feingold cloud–rain system

In this paper, we study the improved aerosol–cloud–precipitation system. Since in our previous article, we guessed that the delayed term y(t−τ) in the denominator tends to cause singularity. So we remove the denominator by time transformation, the original cloud–rain model is transformed into a simpler polynomial system under certain conditions, namely, the improved Koren–Feingold cloud–rain system. Fortunately, the new model meets our expectations. The main work of the present paper is to study the mechanism of cloud and rain interaction by double Hopf bifurcation analysis via the methods of multiple scales. We find some complicated dynamical behaviors of the system via analytical method, such as stable quasi-periodic solutions and chaos. The numerical simulations agree well with those of theoretical analysis, which shows that we are improving the model in the right direction, meanwhile perhaps it suggests that the MMS employed in this paper provides us an effective and accurate method of analyzing double Hopf bifurcation. The above complex dynamical phenomena are of great significance for the researchers to understand the mechanism of aerosol–cloud–precipitation system. And it is also significant to ecosystem and climate control.

Asymptotic stability of quasi-periodic orbits

Stability is one of the most fundamental properties of solutions to differential equations and, for example, can explain the presence of harmful vibrations. While the stability type of fixed points and periodic orbits can be assessed with standard tools, no general and reliable method allows to conclude about the stability of quasi-periodic orbits. Here, we overcome this limitation and develop a novel and universal method to rigorously assess the asymptotic stability of quasi-periodic solutions to differential equations. Furthermore, we develop an automated algorithm for such stability investigations and demonstrate its applicability on two explicit mechanical examples.

The Navier\u2013Stokes Equation with Time Quasi-Periodic External Force: Existence and Stability of Quasi-Periodic Solutions

We prove the existence of small amplitude, time-quasi-periodic solutions (invariant tori) for the incompressible Navier–Stokes equation on the d-dimensional torus mathbb T^d, with a small, quasi-periodic in time external force. We also show that they are orbitally and asymptotically stable in H^s (for s large enough). More precisely, for any initial datum which is close to the invariant torus, there exists a unique global in time solution which stays close to the invariant torus for all times. Moreover, the solution converges asymptotically to the invariant torus for t rightarrow + infty , with an exponential rate of convergence O( e^{- alpha t }) for any arbitrary alpha in (0, 1).

Stability and bifurcations in oscillations of composite laminates with curvilinear fibres under a supersonic airflow

Nonlinear flutter of variable stiffness composite plates—using a reduced-order model benefiting from a large-enough number of modes or degrees of freedom—is the subject of this research. Dynamic of such plates is enriched by different scenarios including period-n (un)stable limit cycle oscillations, quasi-periodic or chaotic solutions, along with Hopf, symmetry breaking and period doubling bifurcations. The fibre path orientation, in any individual ply of these rectangular plates, changes linearly from the left edge to the right one. The plates are subjected to a supersonic airflow. The plate’s displacement field is modelled using a third-order shear deformation theory; a p-version finite element is used to discretize the displacement components. Aerodynamic pressure due to the airflow is approximated using linear piston theory. The self-exciting vibrational full-order model (FOM) is formed by employing the principle of virtual work, and then, the ROM is extracted from the FOM using two reduction techniques, namely static condensation and modal summation method. The inclusion of a large-enough number of modes of vibration in vacuum in the latter technique is critical for the observation of, often overlooked, complex dynamic behaviour of these plates. The equations of motion representing the ROM are solved using the shooting method (to obtain stable and unstable LCOs) and using Newmark method (for stable or non-oscillatory solutions). The rich dynamics of these plates are explored with the help of vibration and flutter mode shapes, phase-plane plots, bifurcation diagrams, fast Fourier transform diagrams and Poincare sections.

Nonstationary coherence-incoherence patterns in nonlocally coupled heterogeneous phase oscillators.

We consider a large ring of nonlocally coupled phase oscillators and show that apart from stationary chimera states, this system also supports nonstationary coherence-incoherence patterns (CIPs). For identical oscillators, these CIPs behave as breathing chimera states and are found in a relatively small parameter region only. It turns out that the stability region of these states enlarges dramatically if a certain amount of spatially uniform heterogeneity (e.g., Lorentzian distribution of natural frequencies) is introduced in the system. In this case, nonstationary CIPs can be studied as stable quasiperiodic solutions of a corresponding mean-field equation, formally describing the infinite system limit. Carrying out direct numerical simulations of the mean-field equation, we find different types of nonstationary CIPs with pulsing and/or alternating chimera-like behavior. Moreover, we reveal a complex bifurcation scenario underlying the transformation of these CIPs into each other. These theoretical predictions are confirmed by numerical simulations of the original coupled oscillator system.

Bifurcations and Chaotic Dynamics of an Axially Accelerating Hinged–Clamped Viscoelastic Beam

This paper investigates stability, bifurcation and dynamic behavior with chaotic response of an axially accelerating hinged–clamped viscoelastic beam under the combined effect of parametric and internal resonant conditions. The beam material obeys the Kelvin–Voigt model in which the material time derivative is used. The axial time-dependent velocity is characterized as a simple harmonic variation about the constant mean speed. The direct method of multiple scales is applied to the nonlinear integro-partial differential equation of motion to reduce it into a set of first-order equations. Here, in particular, aim is fixed to examine the system response in the close proximity of a 3:1 internal resonance between first two transverse modes. Specifically, the fixed point responses are obtained using continuation technique and the dynamic solution is obtained through direct time integration. The steady-state solution has both trivial and two-mode nontrivial solutions with hardening type of nonlinear characteristics. The system displays various bifurcations including super and subcritical pitchfork, Hopf and saddle node bifurcations. The multiple equilibrium solutions get transition to instability of both a saddle node type and a Hopf bifurcation. The damping has different effects on different modes. With decrease in external damping, the maximum amplitude of directly excited first mode is nearly unaffected, but the maximum amplitude of the indirectly excited second mode grows. This shows the energy exchange between the involved modes is encouraged due the presence of internal resonance. Internal damping has stabilizing effect on the transverse modes in the Hopf bifurcation region of the frequency and amplitude response equilibrium curves. It suppresses the nonlinear characteristics, viz. the number of Pitchfork, saddle node and Hopf bifurcation points even to zero for certain cases. Most importantly, multiple nontrivial equilibrium solution curves reduce to two in case of frequency response analysis and a single isolated closed-loop nontrivial solution in amplitude response analysis. Dynamic solution reveals the existence of not only stable periodic, mixed mode and quasiperiodic solutions, but also unstable chaotic oscillations depending on the system parameters and initial conditions in the neighborhoods of Hopf bifurcation regions. The system displays a wide array of interesting dynamic results equipped with various nonlinear features which are not present in the available literature on nonlinear traveling systems.

Quasiperiodicity and Chaos Through Hopf–Hopf Bifurcation in Minimal Ring Neural Oscillators Due to a Single Shortcut

Quasiperiodicity and chaos in a ring of unidirectionally coupled sigmoidal neurons (a ring neural oscillator) caused by a single shortcut is examined. A codimension-two Hopf–Hopf bifurcation for two periodic solutions exists in a ring of six neurons without self-couplings and in a ring of four neurons with self-couplings in the presence of a shortcut at specific locations. The locus of the Neimark–Sacker bifurcation of the periodic solution emanates from the Hopf–Hopf bifurcation point and a stable quasiperiodic solution is generated. Arnold’s tongues emanate from the locus of the Neimark–Sacker bifurcation, and multiple chaotic oscillations are generated through period-doubling cascades of periodic solutions in the Arnold’s tongues. Further, such chaotic irregular oscillations due to a single shortcut are also observed in propagating oscillations in a ring of Bonhoeffer–van der Pol (BVP) neurons coupled unidirectionally by slow synapses.

Double Hopf bifurcation of differential equation with linearly state-dependent delays via MMS

In this paper the dynamics of differential equation with two linearly state-dependent delays is considered, with particular attention focused on non-resonant double Hopf bifurcation. Firstly, we identify the sufficient and necessary conditions of the double Hopf bifurcation by formal linearization, linear stability analysis and Hopf bifurcation theorem. Secondly, in the first time the method of multiple scales (MMS) is employed to classify the dynamics in the neighborhoods of two kinds of non-resonant double Hopf bifurcation points, i.e. Cases III and Ib, in the bifurcation parameters plane. Finally, numerical simulation is executed qualitatively to verify the previously analytical results and demonstrate the rich phenomena, including the stable equilibrium, stable periodic solutions, bistability and stable quasi-periodic solution and so on. It also implies that MMS is simple, effective and correct in higher co-dimensional bifurcation analysis of the state-dependent DDEs. Besides, the other complicated dynamics, such as the switch between torus and phase-locked solution, period-doubling and the route of the break of torus to chaos are also found in this paper.

Fold-Pitchfork Bifurcation, Arnold Tongues and Multiple Chaotic Attractors in a Minimal Network of Three Sigmoidal Neurons

Bifurcations and chaos in a network of three identical sigmoidal neurons are examined. The network consists of a two-neuron oscillator of the Wilson–Cowan type and an additional third neuron, which has a simpler structure than chaotic neural networks in the previous studies. A codimension-two fold-pitchfork bifurcation connecting two periodic solutions exists, which is accompanied by the Neimark–Sacker bifurcation. A stable quasiperiodic solution is generated and Arnold’s tongues emanate from the locus of the Neimark–Sacker bifurcation in a two-dimensional parameter space. The merging, splitting and crossing of the Arnold tongues are observed. Further, multiple chaotic attractors are generated through cascades of period-doubling bifurcations of periodic solutions in the Arnold tongues. The chaotic attractors grow and are destroyed through crises. Transient chaos and crisis-induced intermittency due to the crises are also observed. These quasiperiodic solutions and chaotic attractors are robust to small asymmetry in the output function of neurons.

KAM Tori for Higher Dimensional Quintic Beam Equation

In this paper, we consider higher dimensional quintic beam equation $$\begin{aligned} u_{tt} +\triangle ^2 u+u+u^5 =0, \end{aligned}$$ with periodic boundary conditions, it is proved that the above equation admits a family of small-amplitude linearly stable quasi-periodic solutions.

Bifurcations of a periodically forced microbial continuous culture model with restrained growth rate.

A three dimensional microbial continuous culture model with a restrained microbial growth rate is studied in this paper. Two types of dilution rates are considered to investigate the dynamic behaviors of the model. For the unforced system, fold bifurcation and Hopf bifurcation are detected, and numerical simulations reveal that the system undergoes degenerate Hopf bifurcation. When the system is periodically forced, bifurcation diagrams for periodic solutions of period-one and period-two are given by researching the Poincaré map, corresponding to different bifurcation cases in the unforced system. Stable and unstable quasiperiodic solutions are obtained by Neimark-Sacker bifurcation with different parameter values. Periodic solutions of various periods can occur or disappear and even change their stability, when the Poincaré map of the forced system undergoes Neimark-Sacker bifurcation, flip bifurcation, and fold bifurcation. Chaotic attractors generated by a cascade of period doublings and some phase portraits are given at last.

Bifurcations in a Seasonally Forced Predator–Prey Model with Generalized Holling Type IV Functional Response

A seasonally forced predator–prey system with generalized Holling type IV functional response is considered in this paper. The influence of seasonal forcing on the system is investigated via numerical bifurcation analysis. Bifurcation diagrams for periodic solutions of periods one and two, containing bifurcation curves of codimension one and bifurcation points of codimension two, are obtained by means of a continuation technique, corresponding to different bifurcation cases of the unforced system illustrated in five bifurcation diagrams. The seasonally forced model exhibits more complex dynamics than the unforced one, such as stable and unstable periodic solutions of various periods, stable and unstable quasiperiodic solutions, and chaotic motions through torus destruction or cascade of period doublings. Finally, some phase portraits and corresponding Poincaré map portraits are given to illustrate these different types of solutions.

Double Hopf bifurcation in active control system with delayed feedback: application to glue dosing processes for particleboard

In this paper, we study dynamics in delayed active control system, with particular attention focused on double Hopf bifurcation. Firstly, we identify the critical values for stability switches, Hopf and double Hopf bifurcations using the method of bifurcation analysis. Secondly, by applying the multiple time scales method, the normal form near the double Hopf bifurcation, as well as classifications of local dynamics is analyzed. Finally, an example, associated with active control of ball valve in glue dosing processes for particleboard, is presented to demonstrate the application of the theoretical results. Numerical simulations, including stable equilibrium, stable periodic solutions and stable quasiperiodic solutions, are presented.

Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations

In this paper we consider a class of fully nonlinear forced and reversible Schrödinger equations and prove existence and stability of quasi-periodic solutions. We use a Nash–Moser algorithm together with a reducibility theorem on the linearized operator in a neighborhood of zero. Due to the presence of the highest order derivatives in the non-linearity the classic KAM-reducibility argument fails and one needs to use a wider class of changes of variables such as diffeomorphisms of the torus and pseudo-differential operators. This procedure automatically produces a change of variables, well defined on the phase space of the equation, which diagonalizes the operator linearized at the solution. This gives the linear stability.

On Lotka--Volterra Equations with Identical Minimal Intrinsic Growth Rate

We investigate the long-run behavior of time-dependent Lotka--Volterra equations $({\rm E}_{\phi}):\frac{dx_i}{dt}=x_i(1+\phi(t)+\sum_{j=1}^na_{ij}x_j)$, $i=1,\ldots,n$, on the positive orthant. It is proved that the system $({\rm E}_{\phi})$ is decomposed into $({\rm E}_0)$ and the logistic equation $({\rm L}):\frac{dg}{dt}=g(1+\phi(t)-g)$ in the sense that $({\rm D}):\Phi(t)=g(t)\Psi(\int_0^tg(s)ds)$, where $\Phi(\cdot)$, $\Psi(\cdot)$, and $g(\cdot)$ are the solutions of $({\rm E}_{\phi})$, $({\rm E}_0)$, and (L), respectively. Suppose that $\phi(t)$ is periodic with the mean value ${\cal M}\{\phi\}>-1$. Then the existence of stable equilibrium, periodic solution, and chaotic motion for $({\rm E}_0)$ implies the existence of stable periodic solution, quasiperiodic solution, and chaotic motion for $({\rm E}_{\phi})$, respectively. The complete dynamical classification for 3-dimensional competitive system $({\rm E}_0)$ is provided in terms of competitive coefficients. There are 37 topological classes, in 34 of which any trajectory converges to an equilibrium. Among the remaining three classes, the first has a heteroclinic cycle attracting all positive points except the ray joining the origin and the positive equilibrium; the second possesses a family of periodic orbits attracting all positive points; the third class, where a family of periodic orbits, an asymptotically stable equilibrium, a heteroclinic cycle, and an invariant noncoordinate plane coexist, is a new type we find for the first time. It is shown that for 3-dimensional periodic competitive system $({\rm E}_{\phi})$, any trajectory for Poincaré mapping tends to a fixed point, a periodic orbit, an invariant minimal closed curve, or a heteroclinic cycle. All their attracting domains, which are all cone-like, are exactly described via (D) and carrying simplex theory. The dynamics can be completely classified into 37 classes corresponding to the autonomous systems. All counterparts hold when $\phi(t)$ is another minimal function, such as a quasiperiodic, almost periodic, or almost automorphic function, via (D), the classification above, and skew-product flow theory.

A KAM algorithm for the resonant non-linear Schrödinger equation

We prove, through a KAM algorithm, the existence of large families of stable and unstable quasi-periodic solutions for the NLS in any number of independent frequencies. The main tools are the existence of a non-degenerate integrable normal form proved in [26,27] and a generalization of the quasi-Töplitz functions defined in [31].

Bifurcation phenomena and control analysis in class-B laser system with delayed feedback

In this paper, we study dynamics in delayed class-B laser system, with particular attention focused on Hopf and double Hopf bifurcations. Firstly, we identify the critical values for stability switches, Hopf and double Hopf bifurcations and derive the normal forms near the Hopf and double Hopf bifurcations critical points. By analyzing local dynamics near bifurcation critical points, we show how the delayed feedback control parameters effect the dynamical behaviors of the system. Furthermore, detailed numerical analysis using MATLAB extends the local bifurcation analysis to a global picture, and stable windows are observed as we change control parameter. Namely, even for parameter values not chosen in the neighborhood of the Hopf bifurcation critical points, two families of stable periodic solutions, which are resulted from Hopf bifurcation, exist in a large region of delay, and they merge into a family of stable and globally existed periodic solutions. Finally, by choosing proper control parameters, numerical simulations, including stable equilibrium, stable periodic solutions and stable quasiperiodic solutions are presented to demonstrate the theoretical results. Therefore, in accordance with above theoretical analysis, reasonable lasers with proper control parameters can be designed in order to achieve various applications.

THE CANTOR MANIFOLD THEOREM WITH SYMMETRY AND APPLICATIONS TO PDEs

In this paper we introduce a new Cantor manifold theorem and then apply it to one new type of one-dimensional (1 d ) beam equations $$u_{tt}+u_{xxxx}+mu-2u^2u_{xx}-2uu_x^2=0, m>0,$$ with periodic boundary conditions. We show that the above equation admits small-amplitude linearly stable quasi-periodic solutions corresponding to finite dimensional invaraint tori of an associated infinite dimensional dynamical system. The proof is based on a partial Birkhoff normal form and an infinite dimensional KAM theorem for Hamiltonians with symmetry (cf. [19]).