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  • 1. Adolfsson, J.
    et al.
    Dankowicz, H.
    Nordmark, Arne B.
    KTH, Superseded Departments, Mechanics.
    3D passive walkers: Finding periodic gaits in the presence of discontinuities2001In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 24, no 2, 205-229 p.Article in journal (Refereed)
    Abstract [en]

    This paper studies repetitive gaits found in a 3D passive walking mechanism descending an inclined plane. By using direct numerical integration and implementing a semi-analytical scheme for stability analysis and root finding, we follow the corresponding limit cycles under parameter variations. The 3D walking model, which is fully described in the paper, contains both force discontinuities and impact-like instantaneous changes of state variables. As a result, the standard use of the variational equations is suitably modified. The problem of finding initial conditions for the 3D walker is solved by starting in an almost planar configuration, making it possible to use parameters and initial conditions found for planar walkers. The walker is gradually transformed into a 3D walker having dynamics in all spatial directions. We present such a parameter variation showing the stability and the amplitude of the hip sway motion. We also show the dependence of gait cycle measurements, such as stride time, stride length, average velocity, and power consumption, on the plane inclination. The paper concludes with a discussion of some ideas on how to extend the present 3D walker using the tools derived in this paper.

  • 2. Alekseev, A.A
    et al.
    Kozlov, Alexander
    KTH, School of Computer Science and Communication (CSC), Computational Biology, CB.
    Shalfeev, V.D
    Chaotic regime and synchronous response in frequency controlled oscillator1994In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 5, no 1, 71-77 p.Article in journal (Refereed)
  • 3.
    Farré, Gerard
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Sardanyes, Josep
    Guillamon, Antoni
    Fontich, Ernest
    Coexistence stability in a four-member hypercycle with error tail through center manifold analysis2017In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 90, no 3, 1873-1883 p.Article in journal (Refereed)
    Abstract [en]

    Establishing the conditions allowing for the stable coexistence in hypercycles has been a subject of intensive research in the past decades. Deterministic, time-continuous models have indicated that, under appropriate parameter values, hypercycles are bistable systems, having two asymptotically stable attractors governing coexistence and extinction of all hypercycle members. The nature of the coexistence attractor is largely determined by the size of the hypercycle. For instance, for two-member hypercycles the coexistence attractor is a stable node. For larger dimensions more complex dynamics appear. Numerical results on so-called elementary hypercycles with and species revealed, respectively, coexistence via strongly and weakly damped oscillations. Stability conditions for these cases have been provided by linear stability and Lyapunov functions. Typically, linear stability analysis of four-member hypercycles indicates two purely imaginary eigenvalues and two negative real eigenvalues. For this case, stability cannot be fully characterized by linearizing near the fixed point. In this letter, we determine the stability of a non-elementary four-member hypercycle which considers exponential and hyperbolic replication terms under mutation giving place to an error tail. Since Lyapunov functions are not available for this case, we use the center manifold theory to rigorously show that the system has a stable coexistence fixed point. Our results also show that this fixed point cannot undergo a Hopf bifurcation, as supported by numerical simulations previously reported.

  • 4.
    Fredriksson, Mats H.
    et al.
    KTH, Superseded Departments, Mechanics.
    Borglund, Dan
    KTH, Superseded Departments, Aeronautical Engineering.
    Nordmark, Arne B.
    KTH, Superseded Departments, Mechanics.
    Experiments on the Onset of Impacting Motion Using a Pipe Conveying Fluid1999In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 19, no 3, 261-271 p.Article in journal (Refereed)
    Abstract [en]

    The transition from stable periodic nonimpacting motion to impacting motion, due to variations of parameters, is observable in a wide range of vibro-impact systems. Recent theoretical studies suggest a possible scenario for this type of transition. A key element in the proposed scenario is fulfilled if the oscillatory motion involved in the transition is born in a supercritical Hopf bifurcation. If the onset of impacting motion is close to the Hopf bifurcation, the impacting motion is likely to be chaotic. A numerical simulation of a system of articulated pipes conveying fluid is used to illuminate the theory. An experimental setup is presented, where a cantilevered pipe conveying fluid is unilaterally constrained. Results from experiments are found to be in good qualitative agreement with the theory.

  • 5. Ibragimov, N. H.
    et al.
    Kolsrud, Torbjörn
    KTH, Superseded Departments, Mathematics.
    Lagrangian approach to evolution equations: Symmetries and conservation laws2004In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 36, no 1, 29-40 p.Article in journal (Refereed)
    Abstract [en]

    We show that one can apply a Lagrangian approach to certain evolution equations by considering them together with their associated equations. Consequently, one can employ Noether's theorem and derive conservation laws from symmetries of coupled systems of evolution equations. We discuss in detail the linear and non-linear heat equations as well as the Burgers equation and obtain new non-local conservation laws for the non-linear heat and the Burgers equations by extending their symmetries to the associated equations. We also provide Lagrangians for non-linear Schrodinger and Korteweg-de Vries type systems.

  • 6.
    Kozlov, Alexander
    et al.
    KTH, School of Computer Science and Communication (CSC), Computational Biology, CB.
    Osipov, G.V
    Shalfeev, V.D
    impulse suppression of chaotic oscillations1996In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 1, 113-120 p.Article in journal (Refereed)
    Abstract [en]

    The methods of nonconstant feedback impulse control of chaos are introduced. the approach is based on the similarity of the return of the dissipative continuoustime systems with one dimensional maps. The metods are illustrated for the chua´s circuit Rössler oscillator, and phase-locked loop system.

  • 7.
    Kozlov, Alexander
    et al.
    KTH, School of Computer Science and Communication (CSC), Computational Biology, CB.
    Shalfeev, V.D
    Controlling chaotic oscillations in delayed phase-locked loop1994In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 2, no 2, 36-48 p.Article in journal (Refereed)
  • 8.
    Nordmark, Arne B.
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Biomechanics.
    Piiroinen, P. T.
    Simulation and stability analysis of impacting systems with complete chattering2009In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 58, no 1-2, 85-106 p.Article in journal (Refereed)
    Abstract [en]

    This paper considers dynamical systems that are derived from mechanical systems with impacts. In particular we will focus on chattering-accumulation of impacts-for which local discontinuity mappings will be derived. We will first show how to use these mappings in simulation schemes, and secondly how the mappings are used to calculate the stability of limit cycles with chattering by solving the first variational equations.

  • 9.
    Sjoberg, M.
    et al.
    KTH, Superseded Departments, Vehicle Engineering.
    Kari, Leif
    KTH, Superseded Departments, Vehicle Engineering.
    Nonlinear isolator dynamics at finite deformations: An effective hyperelastic, fractional derivative, generalized friction model2003In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 33, no 3, 323-336 p.Article in journal (Refereed)
    Abstract [en]

    In presenting a nonlinear dynamic model of a rubber vibration isolator, the quasistatic and dynamic motion influences on the force response are investigated within the time and frequency domain. It is found that the dynamic stiffness at the frequency of a harmonic displacement excitation, superimposed upon the long term isolator response, is strongly dependent on static precompression, dynamic amplitude and frequency. The problems of simultaneously modelling the elastic, viscoelastic and friction forces are removed by additively splitting them, modelling the elastic force response by a nonlinear, shape factor based approach, displaying results that agree with those of a neo-Hookean hyperelastic isolator at a long term precompression. The viscoelastic force is modeled by a fractional derivative element, while the friction force governs from a generalized friction element displaying a smoothed Coulomb force. A harmonic displacement excitation is shown to result in a force response containing the excitation frequency and its every other higher-order harmonic, while using a linearized elastic force response model, whereas all higher-order harmonics are present for the fully nonlinear case. It is furthermore found that the dynamic stiffness magnitude increases with static precompression and frequency, while decreasing with dynamic excitation amplitude-eventually increasing at the highest amplitudes due to nonlinear elastic effects-with its loss angle displaying a maximum at an intermediate amplitude. Finally, the dynamic stiffness at a static precompression, using a linearized elastic force response model, is shown to agree with the fully nonlinear model except at the highest dynamic amplitudes.

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