This article introduces a mathematical model that utilizes a nonlinear differential equation to study a range of phenomena such as nonlinear wave processes, and beam deflections. Solving this equation is challenging due to the presence of moving singular points. The article addresses two main problems: first, it establishes the existence and uniqueness of the solution of the equation and, second, it provides precise criteria for determining the existence of a moving singular point. Additionally, the article presents estimates of the error in the analytical approximate solution and validates the results through a numerical experiment.
{"title":"A Study of a Mathematical Model with a Movable Singular Point in a Fourth-Order Nonlinear Differential Equation","authors":"M. V. Gasanov, A. G. Gulkanov","doi":"10.20537/nd230904","DOIUrl":"https://doi.org/10.20537/nd230904","url":null,"abstract":"This article introduces a mathematical model that utilizes a nonlinear differential equation to study a range of phenomena such as nonlinear wave processes, and beam deflections. Solving this equation is challenging due to the presence of moving singular points. The article addresses two main problems: first, it establishes the existence and uniqueness of the solution of the equation and, second, it provides precise criteria for determining the existence of a moving singular point. Additionally, the article presents estimates of the error in the analytical approximate solution and validates the results through a numerical experiment.","PeriodicalId":36803,"journal":{"name":"Russian Journal of Nonlinear Dynamics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135649279","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
K. Suddalai Kannan, S. M. Abdul Kader, V. Chinnathambi, M.V. Sethu Meenakshi, S. Rajasekar
This study examines the phenomenon of vibrational resonance (VR) in a classical positiondependent mass (PDM) system characterized by three types of single-well potentials. These potentials are influenced by an amplitude-modulated (AM) signal with $Omegaggomega$. Our analysis is limited to the following parametric choices: (i) $omega_0^2$, $beta$, $m_0$, $lambda>0$ (type-1 single-well), (ii) $omega_0^2>0$, $beta <0$, $2< m_0 <3$, $1< lambda <2$ (type-2 single-well), (iii) $omega_0^2>0$, $beta <0$, $0< m_0 <2$, $0 The system presents an intriguing scenario in which the PDM function significantly contributes to the occurrence of VR. In addition to the analytical derivation of the equation for slow motions of the system based on the high-frequency signal’s parameters using the method of direct separation of motion, numerical evidence is presented for VR and its basic dynamical behaviors are investigated. Based on the findings presented in this paper, the weak low-frequency signal within the single-well PDM system can be either attenuated or amplified by manipulating PDM parameters, such as mass amplitude ($m_0$) and mass spatial nonlinearity $lambda$. The outcomes of the analytical investigations are validated and further supported through numerical simulations.
{"title":"Nonlinear Resonance in a Position-Dependent Mass-Duffing Oscillator System with Monostable Potentials Driven by an Amplitude-Modulated Signal","authors":"K. Suddalai Kannan, S. M. Abdul Kader, V. Chinnathambi, M.V. Sethu Meenakshi, S. Rajasekar","doi":"10.20537/nd230903","DOIUrl":"https://doi.org/10.20537/nd230903","url":null,"abstract":"This study examines the phenomenon of vibrational resonance (VR) in a classical positiondependent mass (PDM) system characterized by three types of single-well potentials. These potentials are influenced by an amplitude-modulated (AM) signal with $Omegaggomega$. Our analysis is limited to the following parametric choices: <br> (i) $omega_0^2$, $beta$, $m_0$, $lambda>0$ (type-1 single-well), <br> (ii) $omega_0^2>0$, $beta <0$, $2< m_0 <3$, $1< lambda <2$ (type-2 single-well), <br> (iii) $omega_0^2>0$, $beta <0$, $0< m_0 <2$, $0<lambda<1$ (type-3 single-well). <br> The system presents an intriguing scenario in which the PDM function significantly contributes to the occurrence of VR. In addition to the analytical derivation of the equation for slow motions of the system based on the high-frequency signal’s parameters using the method of direct separation of motion, numerical evidence is presented for VR and its basic dynamical behaviors are investigated. Based on the findings presented in this paper, the weak low-frequency signal within the single-well PDM system can be either attenuated or amplified by manipulating PDM parameters, such as mass amplitude ($m_0$) and mass spatial nonlinearity $lambda$. The outcomes of the analytical investigations are validated and further supported through numerical simulations.","PeriodicalId":36803,"journal":{"name":"Russian Journal of Nonlinear Dynamics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135650378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present paper, nonsingular Morse – Smale flows on closed orientable 3-manifolds are considered under the assumption that among the periodic orbits of the flow there is only one saddle and that it is twisted. An exhaustive description of the topology of such manifolds is obtained. Namely, it is established that any manifold admitting such flows is either a lens space or a connected sum of a lens space with a projective space, or Seifert manifolds with a base homeomorphic to a sphere and three singular fibers. Since the latter are prime manifolds, the result obtained refutes the claim that, among prime manifolds, the flows considered admit only lens spaces.
{"title":"Topology of Ambient 3-Manifolds of Non-Singular Flows with Twisted Saddle Orbit","authors":"O. V. Pochinka, D. D. Shubin","doi":"10.20537/nd230905","DOIUrl":"https://doi.org/10.20537/nd230905","url":null,"abstract":"In the present paper, nonsingular Morse – Smale flows on closed orientable 3-manifolds are considered under the assumption that among the periodic orbits of the flow there is only one saddle and that it is twisted. An exhaustive description of the topology of such manifolds is obtained. Namely, it is established that any manifold admitting such flows is either a lens space or a connected sum of a lens space with a projective space, or Seifert manifolds with a base homeomorphic to a sphere and three singular fibers. Since the latter are prime manifolds, the result obtained refutes the claim that, among prime manifolds, the flows considered admit only lens spaces.","PeriodicalId":36803,"journal":{"name":"Russian Journal of Nonlinear Dynamics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135840791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A sufficient condition for the existence of forced oscillations in nonautonomous systems on a plane is presented under the assumption that the magnitude of the nonautonomous perturbation is small. An advantage of the results presented over analytic methods is that they can be applied in degenerate systems as well.
{"title":"A Note on Forced Oscillations in Systems on a Plane","authors":"I. Yu. Polekhin","doi":"10.20537/nd230906","DOIUrl":"https://doi.org/10.20537/nd230906","url":null,"abstract":"A sufficient condition for the existence of forced oscillations in nonautonomous systems on a plane is presented under the assumption that the magnitude of the nonautonomous perturbation is small. An advantage of the results presented over analytic methods is that they can be applied in degenerate systems as well.","PeriodicalId":36803,"journal":{"name":"Russian Journal of Nonlinear Dynamics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135840489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, an adaptive compensator for unknown external disturbances for an inverted pendulum based on the internal model principle is designed. The inverted pendulum is a typical system that has many applications in social life, such as missile launchers, pendubots, human walking and segways, and so on. Furthermore, the inverted pendulum is a high-order nonlinear system, and its parameters are difficult to determine accurately. The physical constraints lead to the complexity of its control design. Besides, there are some unknown external disturbances that affect the inverted pendulum when it operates. The designed adaptive compensation ensures the outputs of the system’s convergence to the desired values while also ensuring a stable system with variable parameters and unknown disturbances. The simulation results are illustrated and compared with the linear quadratic regulator (LQR) controller to show the effectiveness of the proposed compensator.
{"title":"Adaptive Compensation for Unknown External Disturbances for an Inverted Pendulum Based on the Internal Model Principle","authors":"H. D. Long, N. A. Dudarenko","doi":"10.20537/nd231101","DOIUrl":"https://doi.org/10.20537/nd231101","url":null,"abstract":"In this paper, an adaptive compensator for unknown external disturbances for an inverted pendulum based on the internal model principle is designed. The inverted pendulum is a typical system that has many applications in social life, such as missile launchers, pendubots, human walking and segways, and so on. Furthermore, the inverted pendulum is a high-order nonlinear system, and its parameters are difficult to determine accurately. The physical constraints lead to the complexity of its control design. Besides, there are some unknown external disturbances that affect the inverted pendulum when it operates. The designed adaptive compensation ensures the outputs of the system’s convergence to the desired values while also ensuring a stable system with variable parameters and unknown disturbances. The simulation results are illustrated and compared with the linear quadratic regulator (LQR) controller to show the effectiveness of the proposed compensator.","PeriodicalId":36803,"journal":{"name":"Russian Journal of Nonlinear Dynamics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135319135","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
F. Meloni, G.M. Nakamura, B. Grammaticos, A.S. Martinez, M. Badoual
The far-reaching consequences of ecological interactions in the dynamics of biological communities remain an intriguing subject. For decades, competition has been a cornerstone in ecological processes, but mounting evidence shows that cooperation does also contribute to the structure of biological communities. Here, we propose a simple deterministic model for the study of the effects of facilitation and competition in the dynamics of such systems. The simultaneous inclusion of both effects produces rich dynamics and captures the context dependence observed in the formation of ecological communities. Our findings reproduce relevant aspects in plant succession and highlight the role of facilitation mechanisms in species coexistence and conservation efforts.
{"title":"Modeling Cooperation and Competition in Biological Communities","authors":"F. Meloni, G.M. Nakamura, B. Grammaticos, A.S. Martinez, M. Badoual","doi":"10.20537/nd230803","DOIUrl":"https://doi.org/10.20537/nd230803","url":null,"abstract":"The far-reaching consequences of ecological interactions in the dynamics of biological communities remain an intriguing subject. For decades, competition has been a cornerstone in ecological processes, but mounting evidence shows that cooperation does also contribute to the structure of biological communities. Here, we propose a simple deterministic model for the study of the effects of facilitation and competition in the dynamics of such systems. The simultaneous inclusion of both effects produces rich dynamics and captures the context dependence observed in the formation of ecological communities. Our findings reproduce relevant aspects in plant succession and highlight the role of facilitation mechanisms in species coexistence and conservation efforts.","PeriodicalId":36803,"journal":{"name":"Russian Journal of Nonlinear Dynamics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135550946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"In memory of Vyacheslav Z. Grines","authors":"","doi":"10.20537/nd230901","DOIUrl":"https://doi.org/10.20537/nd230901","url":null,"abstract":"<jats:p />","PeriodicalId":36803,"journal":{"name":"Russian Journal of Nonlinear Dynamics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135494876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Developed turbulent flows in which the intervention of external forces is fundamentally important at scales where the inertial range should exist are quite common. Then the cascade processes are not conservative any more and, therefore, it is necessary to adequately describe the external forces acting in the whole range of scales. If the work of these forces has a power law scaling, then one can assume that the integral of motion changes and the preserving value becomes a quadratic quantity, which includes the dependence on the scale. We develop this idea within the framework of shell models of turbulence. We show that, in terms of nonconservative cascades, one can describe various situations, including (as a particular case) the Obukhov – Bolgiano scaling proposed for turbulence in a stratified medium and for helical turbulence with a helicity injection distributed along the spectrum.
{"title":"Nonconservative Cascades in a Shell Model of Turbulence","authors":"P. Frick, A. Shestakov","doi":"10.20537/nd230902","DOIUrl":"https://doi.org/10.20537/nd230902","url":null,"abstract":"Developed turbulent flows in which the intervention of external forces is fundamentally important at scales where the inertial range should exist are quite common. Then the cascade processes are not conservative any more and, therefore, it is necessary to adequately describe the external forces acting in the whole range of scales. If the work of these forces has a power law scaling, then one can assume that the integral of motion changes and the preserving value becomes a quadratic quantity, which includes the dependence on the scale. We develop this idea within the framework of shell models of turbulence. We show that, in terms of nonconservative cascades, one can describe various situations, including (as a particular case) the Obukhov – Bolgiano scaling proposed for turbulence in a stratified medium and for helical turbulence with a helicity injection distributed along the spectrum.","PeriodicalId":36803,"journal":{"name":"Russian Journal of Nonlinear Dynamics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135596197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Memories of Professor Leonid Manevitch","authors":"","doi":"10.20537/nd230907","DOIUrl":"https://doi.org/10.20537/nd230907","url":null,"abstract":"<jats:p />","PeriodicalId":36803,"journal":{"name":"Russian Journal of Nonlinear Dynamics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135840793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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