# Nonlinear Dynamics

## List of Papers (Total 151)

#### Pragmatical adaptive synchronization of different orders chaotic systems with all uncertain parameters via nonlinear control

A new adaptive synchronization scheme by pragmatical asymptotically stability theorem is proposed in this paper. Based on this theorem and nonlinear control theory, a new adaptive synchronization scheme to design controllers can be obtained and especially the constraints for minimum values of feedback gain K in controllers can be derived. This new strategy shows that the constraint ...

#### Computational dynamics of a 3D elastic string pendulum attached to a rigid body and an inertially fixed reel mechanism

A high fidelity model is developed for an elastic string pendulum, one end of which is attached to a rigid body while the other end is attached to an inertially fixed reel mechanism which allows the unstretched length of the string to be dynamically varied. The string is assumed to have distributed mass and elasticity that permits axial deformations. The rigid body is attached to ...

#### On asymptotic approximations of first integrals for second order difference equations

In this paper, the concept of invariance factors for second order difference equations to obtain first integrals or invariants will be presented. It will be shown that all invariance factors have to satisfy a functional equation. Van Horssen (J. Indones. Math. Soc. 13:1–15, 2007) developed a perturbation method for a single first order difference equation based on invariance ...

#### Proportional and derivative control for steady-state vibration mitigation in a piecewise linear beam system

A control using Proportional and/or Derivative feedback (PD-control) is applied on a piecewise linear beam system with a flushing one-sided spring element for steady-state vibration amplitude mitigation. Two control objectives are formulated: (1) minimize the transversal vibration amplitude of the midpoint of the beam at the frequency where the first harmonic resonance occurs, (2) ...

#### Weakly coupled parametrically forced oscillator networks: existence, stability, and symmetry of solutions

In this paper, we discuss existence, stability, and symmetry of solutions for networks of parametrically forced oscillators. We consider a nonlinear oscillator model with strong 2:1 resonance via parametric excitation. For uncoupled systems, the 2:1 resonance property results in sets of solutions that we classify using a combinatorial approach. The symmetry properties for solution ...

#### Time-delayed feedback control of dynamical small-world networks at Hopf bifurcation

We show that time-delayed feedback methods, which have successfully been used to control unstable steady states or periodic orbits, provide a tool to control Hopf bifurcation for a small-world network model with nonlinear interactions and time delays. We choose the interaction strength parameter as a bifurcation parameter. Without control, bifurcation will occur early; meanwhile, ...

#### On the multiple scales perturbation method for difference equations

In the classical multiple scales perturbation method for ordinary difference equations (O Δ Es) as developed in 1977 by Hoppensteadt and Miranker, difference equations (describing the slow dynamics of the problem) are replaced at a certain moment in the perturbation procedure by ordinary differential equations (ODEs). Taking into account the possibly different behavior of the ...

#### Dynamics, chaos and synchronization of self-sustained electromechanical systems with clamped-free flexible arm

An electromechanical system with flexible arm is considered. The mechanical part is a linear flexible beam and the electrical part is a nonlinear self-sustained oscillator. Oscillatory solutions are obtained using an averaging method. Chaotic behavior is studied via the Lyapunov exponent. The synchronization of regular and chaotic states of two such devices is discussed and the ...

#### Singular perturbation methods for slow–fast dynamics

Recently, geometric singular perturbation theory has been extended considerably while at the same time producing many new applications. We will review a number of aspects relevant to non-linear dynamics to apply this to periodic solutions within slow manifolds and to review a number of non-hyperbolic cases. The results are illustrated by examples.

#### Nonlinear dynamic analysis of a structure with a friction-based seismic base isolation system

Many dynamical systems are subject to some form of non-smooth or discontinuous nonlinearity. One eminent example of such a nonlinearity is friction. This is caused by the fact that friction always opposes the direction of movement, thus changing sign when the sliding velocity changes sign. In this paper, a structure with friction-based seismic base isolation is regarded. Seismic ...

#### Interaction between torsional and lateral vibrations in flexible rotor systems with discontinuous friction

In this paper, we analyze the interaction between friction-induced vibrations and self-sustained lateral vibrations caused by a mass-unbalance in an experimental rotor dynamic setup. This study is performed on the level of both numerical and experimental bifurcation analyses. Numerical analyses show that two types of torsional vibrations can appear: friction-induced torsional ...

#### Geometric Proof of Lie's Linearization Theorem

In 1883, S. Lie found the general form of all second-order ordinary differential equations transformable to the linear equation by a change of variables and proved that their solution reduces to integration of a linear third-order ordinary differential equation. He showed that the linearizable equations are at most cubic in the first-order derivative and described a general ...