# Nonlinear Dynamics

## List of Papers (Total 131)

#### The planning of optimal motions of non-holonomic systems

A new method to the planning of optimal motions of the non-holonomic systems is presented. It is based on a non-classical formulation of the Pontryagin Maximum Principle given in variational form, which handles efficiently various control and/or state-dependent constraints. They arise naturally due to both physical limits of the actuators of the non-holonomic systems and potential ...

#### Collisions, mutual losses and annihilation of pulses in a modular nonlinear medium

One of the most important sections of nonlinear wave theory is related to the collisions of single pulses. These often exhibit corpuscular properties. For example, it is well known that solitons described by the Korteweg–de Vries equation and a few other conservative model equations exhibit properties of elastic particles, while shock waves described by dissipative models like ...

#### Analysis of a remarkable singularity in a nonlinear DDE

We investigate the dynamics of the nonlinear DDE (delay-differential equation) $\frac{\hbox {d}^2x}{\hbox {d}t^2}(t)+x(t-T)+x(t)^3=0$, where T is the delay. For $T=0$, this system is conservative and exhibits no limit cycles. For $T>0$, no matter how small T is, an infinite number of limit cycles exist, their amplitudes going to infinity in the limit as T approaches zero. We ...

#### Stability of a convex order one periodic solution of unilateral asymptotic type

In this paper, we consider semi-continuous dynamical systems with linear impulsive conditions, which have a convex order one periodic solution of unilateral asymptotic type. By constructing a sequence of switched systems and using the square approximation of the order one periodic solution, some stability criteria of the order one periodic solution are obtained. Compared with the ...

#### A robust fixed point transformation-based approach for type 1 diabetes control

Modeling and control of diabetes mellitus (DM) are difficult due to the highly nonlinear attitude, time-delay effects, the impulse kind input signals and the lack of continuously available blood glucose (BG) level to be regulated. Regarding the mentioned problems, identification of DM model is crucial. Furthermore, due to the lack of information about the internal states (which ...

#### A new equation and exact solutions describing focal fields in media with modular nonlinearity

Brand-new equations which can be regarded as modifications of Khokhlov–Zabolotskaya–Kuznetsov or Ostrovsky–Vakhnenko equations are suggested. These equations are quite general in that they describe the nonlinear wave dynamics in media with modular nonlinearity. Such media exist among composites, meta-materials, inhomogeneous and multiphase systems. These new models are interesting ...

#### Bifurcation analysis of 4-axle rail vehicle models in a curved track

The article presents authors’ recent results on nonlinear lateral stability of rail vehicles in a curved track. The theories of self-exciting vibrations and bifurcation are the key elements here. The general objective is presentation of extended use of the earlier worked out authors’ method to more complex rail vehicle models. Two 4-axle vehicle models were created. The first one ...

#### A comparative study of the vibro-impact capsule systems with one-sided and two-sided constraints

This paper studies the dynamics of the vibro-impact capsule systems with one-sided and two-sided soft constraints under variations of various system and control parameters, including mass ratio, stiffness ratio, gap of contact, and amplitude and frequency of external excitation. The aim of this study is to optimise the progression speed and energy consumption of the capsule and ...

#### Assessment of predictive control performance using fractal measures

This paper presents novel approach to the task of control performance assessment. Proposed approach does not require any a priori knowledge on process model and uses control error time series data using nonlinear dynamical fractal persistence measures. Notion of the rescaled range R/S plots with estimation of Hurst exponent is applied. Crossover phenomenon is observed in data being ...

#### Classical robots perturbed by Lévy processes: analysis and Lévy disturbance rejection methods

The stability and convergence of state, disturbance and parametric estimates of a robot have been analyzed using the Lyapunov method in the existing literature. In this paper, we analyze the problem of stochastic stability and also prove some results regarding behavior of statistically averaged Lyapunov energy function in the presence of jerk noise modeled as the sum of independent ...

#### Dynamical behavior and exact solution in invariant manifold for a septic derivative nonlinear Schrödinger equation

In this paper, we consider a pulse dynamics in nonlinear optics (fiber-optic communications) in the presence of both self-steepening and septic nonlinear effects. Propagating profiles of the septic derivative nonlinear Schrödinger model which are isolated via coupled integrable invariants of motion, that admits exact solution, are investigated by a method of dynamical systems. By ...

#### Application of dynamic optimisation to stabilise bending moments and top tension forces in risers

The study discusses the problem of determining vertical displacements of a riser’s ends, which, despite its horizontal displacements induced by waves, mitigate stresses. A spatial model of riser dynamics is presented that considers the geometric nonlinearity due to large deflections. The Rigid Finite Element Method was used for riser discretisation. Analyses are reported that ...

#### Inertia forces and shape integrals in the floating frame of reference formulation

Modeling and analysis of complex dynamical systems can be effectively performed using multibody system (MBS) simulation software. Many modern MBS packages are able to efficiently and reliably handle rigid and flexible bodies, often offering a wide choice of different formulations. Despite many advances in modeling of flexible systems, the most widely used formulation remains the ...

#### Egalitarian versus prioritarian approach in multiple task motion planning for nonholonomic systems

In this paper, two different concepts of multiple task motion planning algorithm for nonholonomic systems are considered. The egalitarian approach treats all the tasks equivalently and tries to solve all the tasks simultaneously. In contrast, the prioritarian approach arranges the tasks with decreasing priorities in such a way that the solution of the lower order task should not ...

#### Controlling multistability in a vibro-impact capsule system

This work concerns the control of multistability in a vibro-impact capsule system driven by a harmonic excitation. The capsule is able to move forward and backward in a rectilinear direction, and the main objective of this work is to control such motion in the presence of multiple coexisting periodic solutions. A position feedback controller is employed in this study, and our ...

#### Effects of play and inerter nonlinearities on the performance of tuned mass damper

In this paper, we analyze the dynamics of tuned mass dampers with inerters. In the beginning, we describe the influence of inertance value with respect to the overall mass of the damping device. For further analysis, we pick three practically significant cases—each corresponding to different composition of tuned mass damper inertia. Then, we focus on the effects caused by different ...

#### Reduction of dimension for nonlinear dynamical systems

We consider reduction of dimension for nonlinear dynamical systems. We demonstrate that in some cases, one can reduce a nonlinear system of equations into a single equation for one of the state variables, and this can be useful for computing the solution when using a variety of analytical approaches. In the case where this reduction is possible, we employ differential elimination ...

#### Enhancing the isolation performance by a nonlinear secondary spring in the Zener model

In order to obtain an isolator with low resonance amplitude as well as good isolation performance at high frequencies, this paper explores the usage of nonlinear stiffness elements to improve the transmissibility efficiency of a sufficient linear damped vibration isolator featured with the Zener model. More specifically, we intend to improve its original poor high-frequency ...

#### Switching processes in polynomiography

Mandelbrot and Julia sets are examples of fractal patterns generated in the complex plane. In the literature, we can find many generalizations of those sets. One of such generalizations is the use of switching process. In this paper, we introduce some switching processes to another type of complex fractals, namely polynomiographs. Polynomiograph is an image presenting the ...