Different Types Of Oscillations Research Articles

Multistable dynamics and chaos in a system consisting of an inertial neuron coupled to a van der Pol oscillator

Abstract We consider a dynamical system consisting of a van der Pol oscillator linearly coupled to an inertial neuron with two wells potential. Analytical studies are conducted focusing on the energy computation, the dissipation and symmetry, as well as the determination and characterization of the equilibrium points. We define the parameter ranges related to different types of oscillations in the coupled system in order to have an overall idea of the nature of the attractors (hidden or self-excited) that may exist. We apply numerical analysis techniques (2-parameter diagrams, bifurcation analysis, phase portraits, basins of attractions, etc) in accordance with the previous operating range in order to shed light on the plethora of competing dynamics of the model and possible forms of strange attractors as well. Another salient point of this work is the coexistence between five self-excited attractors (limit cycle and chaos) with a hidden attractor (limit cycle). We also examine the impact of symmetry breaking on the system response. An appropriate analog simulator of the coupled system is designed and simulated in PSpice in order to check the results reported during the theoretical analyses. We believe that the results of the present work complement and enrich previously published ones concerning the dynamics of a system composed of a van der pol oscillator coupled to a (non-oscillating) double-well oscillator.

Bifurcation analysis of a Parkinson’s disease model with two time delays

In this paper, a cortex-basal ganglia model about Parkinson’s disease with two time delays is studied, and the critical conditions for Hopf bifurcation are derived. The results show that time delays can change the state of basal ganglia. The basal ganglia is stable when the delays are small. However, when the time delay is greater than the corresponding bifurcation critical point, different types of oscillations occur in the basal ganglia. The larger the time delays, the more active the neuronal clusters in the basal ganglia. Furthermore, the bidirectional Hopf bifurcation is found by studying the connection weights between different neural nuclei. Finally, the influence of connection weight and time delay which are related to the internal segment of the globus pallidus on its oscillation is discussed. Research shows that reducing the connection weight and the corresponding time delay in excitatory neuronal clusters, or increasing the connection weight and decreasing the corresponding time delay in inhibitory neuronal clusters, can improve the oscillation of Parkinson’s disease.

A nested bistable module within a negative feedback loop ensures different types of oscillations in signaling systems

In this article, we consider a double phosphorylation cycle, a ubiquitous signaling component, having the ability to display bistability, a behavior strongly related to the existence of positive feedback loops. If this component is connected to other signaling elements, it very likely undergoes some sort of protein–protein interaction. In several cases, these interactions result in a non-explicit negative feedback effect, leading to interlinked positive and negative feedbacks. This combination was studied in the literature as a way to generate relaxation-type oscillations. Here, we show that the two feedbacks together ensure two types of oscillations, the relaxation-type ones and a smoother type of oscillations functioning in a very narrow range of frequencies, in such a way that outside that range, the amplitude of the oscillations is severely compromised. Even more, we show that the two feedbacks are essential for both oscillatory types to emerge, and it is their hierarchy what determines the type of oscillation at work. We used bifurcation analyses and amplitude vs. frequency curves to characterize and classify the oscillations. We also applied the same ideas to another simple model, with the goal of generalizing what we learned from signaling models. The results obtained display the wealth of oscillatory dynamics that exists in a system with a bistable module nested within a negative feedback loop, showing how to transition between different types of oscillations and other dynamical behaviors such as excitability. Our work provides a framework for the study of other oscillatory systems based on bistable modules, from simple two-component models to more complex examples like the MAPK cascade and experimental cases like cell cycle oscillators.

Experimental study on the passive regulation of a CO2 two-phase thermosyphon loop

• The passive regulation mechanism of CO 2 TPTL was revealed. • Three typical operation states of the TPTL occurred as the heat load increased. • Each operation state showed different characteristics as the filling ratio changed. • The largest normal operation heat load range was found under 45% filling ratio. Vapor–liquid distribution and flow characteristics in a two-phase thermosyphon loop usually exhibit regular variation under varying working conditions, which can be referred to as passive regulation. Currently, the passive regulation law of a CO 2 two-phase thermosyphon loop has not been well understood. In this study, experiments were performed to examine the passive regulation mechanism of a CO 2 two-phase thermosyphon loop under different heat loads and filling ratios. Local flow visualization was adopted, combined with the measurement of operating parameters. The CO 2 two-phase thermosyphon loop entered the pre-start, oscillatory, and stable operation stages successively with the increase in heat load. Vapor–liquid counterflow was observed in the downcomer in the pre-start stage. Temperature change due to thermal unbalance between the vapor and liquid can be observed along the riser or the downcomer. In the oscillatory stage, different types of oscillation were observed under different filling ratios. In the stable stage, the two-phase thermosyphon loop worked under good conditions until the heat transfer limit was reached. The two-phase thermosyphon loop reached its maximum heat transfer limit of 1200 W at a filling ratio of 45%. The normal operation heat load range of the two-phase thermosyphon loop was 380–650 W, 320–770 W, 700–1100 W, 600–1200 W, and 400–630 W for filling ratios of 20%, 25%, 40%, 45%, and 50%, respectively.

Linear response for pseudo-Hermitian Hamiltonian systems: Application to PT -symmetric qubits

Motivated by the recent advances in modelling the pseudo-Hermitian Hamiltonian (pHH) systems using superconducting qubits we analyze their quantum dynamics subject to a small time-dependent perturbation. In particular, We develop the linear response theory formulation suitable for application to various pHH systems and compare it to the ones available in the literature. We derive analytical expressions for the generalized temporal quantum-mechanical correlation function $C(t)$ and the time-dependent dynamic susceptibility $\chi(t) \propto \text{Im} ~C(t)$. We apply our results to two \textit{PT}-symmetric non-Hermitian quantum systems: a single qubit and two unbiased/biased qubits coupled by the exchange interaction. For both systems we obtain the eigenvalues and eigenfunctions of the Hamiltonian, identify \textit{PT}-symmetry unbroken and broken quantum phases and quantum phase transitions between them. The temporal oscillations of the dynamic susceptibility of the qubits polarization ($z$-projection of the total spin), $\chi(t)$, relate to {\it ac} induced transitions between different eigenstates and we analyze the dependencies of the oscillations frequency and the amplitude on the gain/loss parameter $\gamma$ and the interaction strength $g$. Studying the time dependence of $\chi(t)$ we observe different types of oscillations, i.e. undamped, heavily damped and amplified ones, related to the transitions between eigenstates with broken (unbroken) $PT$-symmetry. These predictions can be verified in the microwave transmission experiments allowing controlled simulation of the pHH systems.

Internal energy exchanges and chaotic dynamics in an intrinsically coupled system

Intrinsically coupled nonlinear systems typically present different oscillating components that exchange energy among themselves. A paradigmatic example is the spring pendulum, for which we identify spring, pendulum, and coupled oscillations. We propose a new approach that properly accounts for the nonlinear coupling, and allows the analysis of energy exchanges among the different types of oscillation. We obtain that the rate of energy exchanges is enhanced for chaotic orbits. Moreover, the highest rates for the coupling occur in the vicinity of the homoclinic tangle of the primary hyperbolic point embedded in a chaotic sea. The results demonstrate a clear relation between internal energy exchanges and the dynamics of coupled systems, being an efficient new way to distinguish regular from chaotic orbits.

Analysis of the onset of a regime shift and detecting early warning signs of major population changes in a two-trophic three-species predator-prey model with long-term transients.

Identifying early warning signs of sudden population changes and mechanisms leading to regime shifts are highly desirable in population biology. In this paper, a two-trophic ecosystem comprising of two species of predators, competing for their common prey, with explicit interference competition is considered. With proper rescaling, the model is portrayed as a singularly perturbed system with fast prey dynamics and slow dynamics of the predators. In a parameter regime near singular Hopf bifurcation, chaotic mixed-mode oscillations (MMOs), featuring concatenation of small and large amplitude oscillations are observed as long-lasting transients before the system approaches its asymptotic state. To analyze the dynamical cause that initiates a large amplitude oscillation in an MMO orbit, the model is reduced to a suitable normal form near the singular-Hopf point. The normal form possesses a separatrix surface that separates two different types of oscillations. A large amplitude oscillation is initiated if a trajectory moves from the "inner" to the "outer side" of this surface. A set of conditions on the normal form variables are obtained to determine whether a trajectory would exhibit another cycle of MMO dynamics before experiencing a regime shift (i.e. approaching its asymptotic state). These conditions serve as early warning signs for a sudden population shift as well as detect the onset of a regime shift in this ecological model.

Potential Oscillations during Electroreduction of Iodate and Nitrate

Electrochemical oscillations are attractive phenomena from the viewpoint of dynamic self-organization of molecular systems. In general, an N-shaped negative differential resistance (N-NDR) plays a crucial role in the appearance of oscillations because it gives rise to oscillatory instability [1]. Most of the oscillations can be classified into an N-NDR type or “hidden” N-NDR (HN-NDR) type oscillator. The former shows current oscillations under potential controlled conditions and hysteresis loops under current-controlled conditions, whereas the latter shows not only current oscillations but also potential oscillations. Thus, the electrochemical system that shows potential oscillations falls into an HN-NDR type oscillator. However, it is not always true because there are different types of oscillators that show potential oscillations, namely, strictly potentiostatic type, S-shaped NDR (S-NDR) type, coupled N-NDR (CN-NDR) type, and electrochemical reactions and diffusion–convection (ERDC) type oscillators.When iodate (IO3 -) is reduced on an Ag, Au, or Pt electrode in alkaline media, potential oscillations appear in conjunction with hydrogen evolution reaction [2-4]. When the potential of an electrode (E) is scanned at a fast rate, an N-shaped relationship between current (I) and E (i.e., a negative slope in an I-E curve) is observed in the potential range of the oscillations [3]. Thus, the oscillations seem to be attributable to N-NDR characteristics and are likely to be an HN-NDR type. It is highly possible that a Frumkin-type coulomb interaction between IO3 - ions within the diffusive double-layer plays an important role in the N-NDR characteristics, as suggested by Strasser [3]. On the other hand, another reasonable explanation for the potential oscillations was proposed by Li [2, 4]. The negative slope can be attributed to the limiting depletion of IO3 - ions and the oscillations are an ERDC type. That is, the alternative diffusion-limited depletion and convection-enhanced replenishment of the ions by hydrogen evolution causes the oscillations. It should be noted that details concerning these two interpretations are described in literature [1, 5].In this present work, to obtain greater insight into the mechanism of the oscillations, we study the potential oscillations using Au, Ag, and Cu electrodes (see Figure 1) and compare them with the oscillations that appear during the reduction of nitrate (NO3 -) ions. The NO3 - reduction on Ag and Cu electrodes shows two types of potential oscillations in the potential region of hydrogen evolution reaction (called oscillations II and III), as we have reported previously [6, 7]. Recent studies revealed that oscillations II and III can be explained in the framework of ERDC and HN-NDR types, respectively. By considering that the iodate system has common features as the nitrate system, it is reasonable that the iodate system also shows N-NDR characteristics in addition to ERDC. This idea is in line with all the previous studies including the results from electrochemical impedance spectroscopy [3], electrochemical quartz crystal microbalance, and in situ Raman spectroscopy [4], which will be discussed in this presentation. REFERENCES M. Orlik, Self-Organization in Electrochemical Systems I, Springer-Verlag Berlin Heidelberg, Berlin (2012). Z. Li, J. Cai and S. Zhou, J. Chem. Soc., Faraday Trans., 93, 3519 (1997). P. Strasser, M. Lübke, C. Eickes and M. Eiswirth, J. Electroanal. Chem., 462, 19 (1999). Z. L. Li, Z. J. Niu, T. H. Wu, H. D. Nie and X. M. Xiao, Electrochem. Commun., 5, 297 (2003). M. Orlik, Self-Organization in Electrochemical Systems II, Springer-Verlag Berlin Heidelberg, Berlin (2012). Y. Mukouyama, S. Yamamoto, R. Nakazato, S. Nakanishi and H. Okamoto, ECS Trans, 58, 61 (2013). Y. Mukouyama, S. Yamamoto, S. Nakanishi and H. Okamoto, ECS Trans, 58, 85 (2014). FIGURE CAPTION Figure 1 I–E curves obtained using (top) an Au-wire electrode, (middle) an Ag-wire electrode, and (bottom) a Cu-wire electrode, under (left) potential-controlled conditions at a scan rate of 10 mV s−1 and (right) current-controlled conditions at a scan rate of 0.05 mA s−1. The electrolyte: 0.1 M NaOH + 0.1 M KIO3. Figure 1

Invariants of optimal integration of rapidly oscillatory functions

The paper presents some common elements (invariants) of optimal integration of rapidly oscillatory functions for the different types of oscillations, in particular, for calculating the Fourier transform from finite functions, wavelet transform, and Bessel transform. Their brief description is given. The application of the invariants allows to increase the potential of quadrature formulas due to the fullest use of apriori information. Invariants form the basis of computer technology of integration of rapidly oscillatory functions with a given accuracy with limited computational resources.

Linear and Non-Linear Middle Infrared Spectra of Penicillin G in the CO Stretching Mode Region

In this work we report the linear and non-linear IR spectral response characterization of the CO bonds of PenicillinG sodium salt in D2O and in DMSO−d6 solutions. In order to better characterize the spectral IR features in the CO stretching region, broadband middle infrared pump-probe spectra are recorded. The role of hydrogen bonds in determining the inhomogeneous broadening and in tuning anharmonicity of the different types of oscillators is exploited. Narrow band pump experiments, at the three central frequencies of β−lactam, amide and carboxylate CO stretching modes, identify the couplings between the different types of CO oscillators opening the possibility to gather structural dynamic information. Our results show that the strongest coupling is between the β−lactam and the carboxylate CO vibrational modes.

Инерционно-циркуляционный принцип плавания и полётагидро-и аэробионтов. Часть 2

For the case of modeling in the nonlinear ideal formulation of oscillations of an infinitely thin profile of the wing-propulsion, three components of the thrust force coefficient were distinguished - inertial, circulating, and vortex. The contribution to the traction force of each of the obtained components is investigated and the mechanisms of wing traction force formation at different types of oscillations are explained. It is revealed that the inertial-circulating principle underlies the work of the wing-propulsion. The inductive effect of the vortex trail on the traction force is small and negative.

Инерционно-циркуляционный принцип плавания и полётагидро-и аэробионтов. Часть 1

For the case of modeling in the nonlinear ideal formulation of oscillations of an infinitely thin profile of the wing-propulsion, three components of the thrust force coefficient were distinguished - inertial, circulating, and vortex. The contribution to the traction force of each of the obtained components is investigated and the mechanisms of wing traction force formation at different types of oscillations are explained. It is revealed that the inertial-circulating principle underlies the work of the wing-propulsion. The inductive effect of the vortex trail on the traction force is small and negative.

Hidden N-Shaped Negative Differential Resistance Induced By Hydrogen Bubble Evolution during Reduction of Nitrate Ions on Cu Electrode

Electrochemical oscillations are attractive phenomena from the viewpoint of dynamic self-organization of molecular systems. In general, an N-shaped negative differential resistance (N-NDR) plays a crucial role in the appearance of oscillations because it gives rise to an oscillatory instability [1]. Most of the oscillations can be classified into an N-NDR type or “hidden” N-NDR (HN-NDR) type oscillator. The former shows current oscillations under potential controlled conditions and hysteresis loops under current controlled conditions, whereas the latter shows not only current oscillations but also potential oscillations. Thus, the electrochemical system that shows potential oscillations falls into an HN-NDR type oscillator. However, it is not always true because there are different types of oscillators that show potential oscillations, namely, strictly potentiostatic type, S-shaped NDR (S-NDR) type, coupled N-NDR (CN-NDR) type, and electrochemical reactions and diffusion–convection (ERDC) type oscillators [2].Electrochemical reduction reactions are accompanied with hydrogen bubble evolution when the electrode potential is much more negative than the onset potential of hydrogen evolution reaction (HER). The bubble evolution often causes a disturbance in the electrochemical measurements and analysis. Thus, electrochemical studies are mostly conducted in the potential region where the HER does not occur or where bubbles do not affect the electrochemical reactions. Probably for this reason, little has been studied on the electrochemical reduction of nitrate ions in the potential region of the HER, though the reduction has been extensively studied for years.We recently found that three types of oscillations (named oscillations I, II, and III) occurred during the reduction on Cu electrode [3, 4]. Interestingly, as shown in Figure 1, they appeared as current and potential oscillations in the potential region of the HER. The appearance of oscillations I and III depended on the acidity of solutions: oscillation I appeared when the nitrate reduction occurred in acidic solutions (the left panels) whereas oscillation III did in basic ones (the right panels). On the other hand, oscillation II appeared irrespective of the acidity.This work focuses on studying the mechanism of oscillation II. A high-speed video recording showed that hydrogen bubbles evolved synchronously with oscillation II. More importantly, electrochemical impedance measurements indicated that oscillation II was an HN-NDR type oscillator (the panels d1 and d2). These led to a clear conclusion that the bubble evolution induced positive and negative feedback effects during the nitrate reduction. Interestingly, a detailed study showed that the potential oscillation that was observed when the solution was acidic (e.g., oscillation II shown in b1 and c1) behaved like an ERCD type oscillator. In the presentation, we will discuss the difference between the HN-NDR and ERDC oscillators and explain the mechanism of oscillation II. Figure 1

The relationship between car following string instability and traffic oscillations in finite-sized platoons and its use in easing congestion via connected and automated vehicles with IDM based controller

This paper focuses on two fundamental issues in traffic flow modelling: string stability of car following (CF), and oscillation of traffic flow. Its aim is to explore the complementary use of CF instability analysis and oscillation analysis to compress and ease traffic congestion in a connected environment. Each of these topics has been extensively investigated in the literature. However, each topic has been investigated separately and, despite the inherent conceptual and empirical closeness of the two concepts, little effort has been devoted to untangling their relationship.To address this failing, we first define four types of oscillation – amplitude-decay oscillation, amplitude-ceiling oscillation, speed-deviation ceiling oscillation, and speed-deviation growth oscillation – and reveal their similarities and dissimilarities to CF instability. Based on the stability criterion, we then develop oscillation criteria to identify different types of oscillation by relaxing two unrealistic assumptions used in CF stability analysis (i.e., infinitely-long platoon and long-wavelength perturbation). Finally, to demonstrate how CF instability analysis and oscillation analysis can be combined to influence individual vehicle stability and improve traffic oscillations in a connected environment, a platoon of vehicles that experience an oscillation in the NGSIM data is used in a case study. In this case study, different control factors are used for different vehicle types: connected (but human-driven) vehicles, and automated vehicles.Our analysis shows that a higher stability of some individual vehicles can alleviate the oscillation severity for the platoon. It also shows that desired time gap and maximum acceleration are two promising parameters that can be used to both improve individual vehicle stability and significantly smooth the oscillation of the platoon in a connected and/or automated environment. Of particular note, when considering all of the factors explored in our analysis, adjustment of the desired time gap is the most effective factor in smoothing traffic oscillations.

Neural-Like Dynamics in a Phase-Locked Loop System with Delayed Feedback

We proposed a model of neural dynamics based on a phase-locked loop system with delayed feedback. The model can demonstrate chaotic oscillation modes, in which switching between qualitatively different types of oscillations that are characteristic of neural activity takes place.

Нейроподобная динамика в системе фазовой автоподстройки частоты с запаздывающей обратной связью

A model of the neural dynamics is proposed based on phase-locked loop system with delayed feedback. The model is able to demonstrate chaotic oscillatory regimes, which have typical for neural activity switching between qualitatively different types of oscillations.

Local control of an array of the globally coupled oscillators

Two control methods for stabilization of the steady states of the globally coupled second-order oscillators via local feedback are described. The first technique is based on the proportional feedback with a constant adjustable reference. The second technique is an adaptive one, employing the first-order stable filter. Three different types of oscillators have been considered: (1) strongly asymmetric version of the FitzHugh–Nagumo oscillators, (2) the classical FitzHugh–Nagumo oscillators, and (3) the van der Pol oscillators. Numerical simulations have been performed for all three types of oscillators. Detailed mathematical analysis and hardware experiments have been carried out with the asymmetric FitzHugh–Nagumo oscillators.

Dynamics and stability of chimera states in two coupled populations of oscillators.

We consider networks formed from two populations of identical oscillators, with uniform strength all-to-all coupling within populations and also between populations with a different strength. Such systems are known to support chimera states in which oscillators within one population are perfectly synchronized while in the other the oscillators are incoherent and have a different mean frequency from those in the synchronous population. Assuming that the oscillators in the incoherent population always lie on a closed smooth curve C, we derive and analyze the dynamics of the shape of C and the probability density on C for four different types of oscillators. We put some previously derived results on a more rigorous footing and analyze two new systems.

Potential Oscillation during Hydrogen Evolution Reaction: Hidden N-Shaped Negative Differential Resistance Due to Hydrogen Bubbles

Electrochemical oscillations are attractive phenomena from the viewpoint of dynamic self-organization of molecular systems. In general, an N-shaped negative differential resistance (N-NDR) plays a crucial role in the appearance of oscillations because it gives rise to an oscillatory instability [1]. Most of the oscillations can be classified into an N-NDR type or “hidden” N-NDR (HN-NDR) type oscillator. The former shows current oscillations under potential controlled conditions and hysteresis loops under current controlled conditions, whereas the latter shows not only current oscillations but also potential oscillations. Thus, the electrochemical system that shows potential oscillations falls into an HN-NDR type oscillator. However, it is not always true because there are different types of oscillators that show potential oscillations, namely, strictly potentiostatic type, S-shaped NDR (S-NDR) type, coupled N-NDR (CN-NDR) type, and electrochemical reactions and diffusion–convection (ERDC) type oscillators. Hydrogen evolution reaction (HER), which is the cathode reaction in water electrolysis, has been extensively studied using various kinds of electrodes because it is a fundamental electrochemical reaction and also because it is an important reaction for the production of H2. When the electrode potential is much negative than the onset potential of HER, a large number of hydrogen bubbles are formed on electrode surface. This often causes a disturbance in electrochemical reactions, leading to difficulties in electrochemical measurements, controls, and analysis. Thus, the HER has been mostly studied in a low overpotential region where no bubbles are formed or in an intermediate region where bubbles do not affect electrochemical measurements. We have reported [2] that the HER on Pt and Au in 0.10 M (M = mol/dm3) H2SO4 solution is anomalously affected by salts such as Na2SO4 and K2SO4 in a high overpotential region, e.g., more negative than -0.5 V vs. SHE, as shown in Figure 1a. The reduction current due to the HER became low (in absolute value) when Na2SO4 or K2SO4 was added to the solution. Interestingly, a potential oscillation, named HER oscillation, appeared under current controlled conditions (Figures 1b and 1c). As reported recently [3], the oscillation also appeared in the absence of the salts when the concentration of H2SO4 was as low as 0.03 M. The amount of hydrogen bubbles oscillated synchronously with the potential oscillation: the bubbles evolved more vigorously in the low potential region where the reduction of water occurred than in the high potential region where the reduction of H+ occurred. Therefore, we proposed that the bubble evolution played role in both the positive and negative feedback mechanisms, the combination of which caused an oscillatory instability. The proposed mechanism, however, cannot explain N-NDR characteristics though the oscillation seems to be an HN-NDR type oscillator because it appears as potential oscillations. In the present study, to clarify how the bubble evolution is related to N-NDR characteristics and an NDR-hiding factor, the bubble evolution is studied using Pt electrodes under atmospheric pressure (0.1 M Pa) conditions and also under high-pressure (up to 1 M Pa) conditions. Furthermore, the bubble behaviour is carefully observed by using a high-speed camera. In this presentation, we will show how hydrogen bubbles concurrently induce and hide the N-NDR characteristics. REFERENCES [1] M. Orlik, Self-Organization in Electrochemical Systems I, Springer, Berlin (2012). [2] Y. Mukouyama, M. Kikuchi, H. Okamoto, J. Electroanal. Chem., 617 (2008) 179. [3] Y. Mukouyama, R. Nakazato, T. Shiono, S. Nakanishi, H. Okamoto, J. Electroanal. Chem., 713 (2014) 39. FIGURE CAPTION Figure 1. Current (I) - potential (E) curves for a Pt electrode in 0.10 M H2SO4 solution with or without 0.05 M Na2SO4, measured (a) under potential controlled conditions and (b) under current controlled conditions. (c) Time course of E measured at I = -16 mA. Figure 1

Active poroelastic two-phase model for the motion of physarum microplasmodia.

The onset of self-organized motion is studied in a poroelastic two-phase model with free boundaries for Physarum microplasmodia (MP). In the model, an active gel phase is assumed to be interpenetrated by a passive fluid phase on small length scales. A feedback loop between calcium kinetics, mechanical deformations, and induced fluid flow gives rise to pattern formation and the establishment of an axis of polarity. Altogether, we find that the calcium kinetics that breaks the conservation of the total calcium concentration in the model and a nonlinear friction between MP and substrate are both necessary ingredients to obtain an oscillatory movement with net motion of the MP. By numerical simulations in one spatial dimension, we find two different types of oscillations with net motion as well as modes with time-periodic or irregular switching of the axis of polarity. The more frequent type of net motion is characterized by mechano-chemical waves traveling from the front towards the rear. The second type is characterized by mechano-chemical waves that appear alternating from the front and the back. While both types exhibit oscillatory forward and backward movement with net motion in each cycle, the trajectory and gel flow pattern of the second type are also similar to recent experimental measurements of peristaltic MP motion. We found moving MPs in extended regions of experimentally accessible parameters, such as length, period and substrate friction strength. Simulations of the model show that the net speed increases with the length, provided that MPs are longer than a critical length of ≈ 120 μm. Both predictions are in line with recent experimental observations.