We present a fully automated method for the optimal state space reconstruction from univariate and multivariate time series. The proposed methodology generalizes the time delay embedding procedure by unifying two promising ideas in a symbiotic fashion. Using non-uniform delays allows the successful reconstruction of systems inheriting different time scales. In contrast to the established methods, the minimization of an appropriate cost function determines the embedding dimension without using a threshold parameter. Moreover, the method is capable of detecting stochastic time series and, thus, can handle noise contaminated input without adjusting parameters. The superiority of the proposed method is shown on some paradigmatic models and experimental data from chaotic chemical oscillators.

/pdf/a-unified-and-automated-approach-to-attractor-reconstruction-1egus89ggd.pdf

A unified and automated approach to attractor reconstruction

Infinitely many coexisting attractors of a dual memristive Shinriki oscillator and its FPGA digital implementation

This paper provides a mass, momentum, and energy flux conservation analysis between the linear and nonlinear steady-state wave groups. Convergent high-order solutions for nonlinear wave groups with multiple steady-state near resonances in deep water have been obtained by means of the homotopy analysis method. The small divisors associated with nearly resonant components are transformed to singularities that are originally caused by exact resonances by a piecewise auxiliary linear operator. Both two primary components and other nearly resonant ones are considered in the initial guess to search for finite amplitude wave groups. It is found that as nonlinearity of wave groups increases, more wave components appear in the spectrum due to the nearly resonant interactions. The nonlinear wave fields change from the initial bi-chromatic waves that contain only two nontrivial primary components into the steady-state resonant waves that contain both two primary components and other nearly resonant ones. The conserv...

Mass, momentum, and energy flux conservation between linear and nonlinear steady-state wave groups

The main purpose of the paper is to reveal the mechanism of certain special phenomena in bursting oscillations such as the sudden increase of the spiking amplitude. When multiple equilibrium points coexist in a dynamical system, several types of stable attractors via different bifurcations from these points may be observed with the variation of parameters, which may interact with each other to form other types of bifurcations. Here we take the modified van der Pol–Duffing system as an example, in which periodic parametric excitation is introduced. When the exciting frequency is far less than the natural frequency, bursting oscillations may appear. By regarding the exciting term as a slow-varying parameter, the number of the equilibrium branches in the fast generalized autonomous subsystem varies from one to five with the variation of the slow-varying parameter. The equilibrium branches may undergo different types of bifurcations, such as Hopf and pitchfork bifurcations. The limit cycles, including the cycles via Hopf bifurcations and the cycles near the homo-clinic orbit, may interact with each other to form the fold limit cycle bifurcations. With the increase of the exciting amplitude, different stable attractors and bifurcations of the generalized autonomous fast subsystem involve the full system, leading to different types of bursting oscillations. Fold limit cycle bifurcations may cause the sudden change of the spiking amplitude, since at the bifurcation points, the trajectory may oscillate according to different stable limit cycles with obviously different amplitudes. At the pitchfork bifurcation point, two possible jumping ways may result in two coexisted asymmetric bursting attractors, which may expand in the phase space to interact with each other to form an enlarged symmetric bursting attractor with doubled period. The inertia of the movement along the stable equilibrium may cause the trajectory to pass across the related bifurcations, leading to the delay effect of the bifurcations. Not only the large exciting amplitude, but also the large value of the exciting frequency may increase inertia of the movement, since in both the two cases, the change rate of the slow-varying parameter may increase. Therefore, a relative small exciting frequency may be taken in order to show the possible influence of all the equilibrium branches and their bifurcations on the dynamics of the full system. 

https://www.researchsquare.com/article/rs-625992/latest.pdf

On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol–Duffing system with slow-varying periodic excitation

In this paper, the limit cycles, period-doubling, and quasi-periodic solutions of the forced Van der Pol oscillator and the forced Van der Pol-Duffing oscillator are studied by combining the homotopy analysis method (HAM) with the multi-scale method analytically. Comparisons of the obtained solutions and numerical results show that this method is effective and convenient even when t is large enough, and the convergence of the approximate solutions is discussed by the so-called discrete square residual error. This method is a capable tool for solving this kind of nonlinear problems.

On the limit cycles, period-doubling, and quasi-periodic solutions of the forced Van der Pol-Duffing oscillator

Abstract In this article, a modified Bretherton equation is considered to further check if steady-state multiple resonances exist not only for water waves but also for other dispersive medium. The linear resonance condition analysis shows that different components may interact with each other so multiple resonances may happen. Convergent steady-state solutions are obtained by solution procedure based on the homotopy analysis method (HAM) and the collocation method. Amplitude spectrum analysis confirms that more components indeed join the resonance as the nonlinearity increases. This article suggests that steady-state multiple resonance may exist in other dispersive system.

Jifeng Cui

Papers

On the limit cycles, period-doubling, and quasi-periodic solutions of the forced Van der Pol-Duffing oscillator

On Steady-State Multiple Resonances for a Modified Bretherton Equation