# Nonlinear Dynamics

## List of Papers (Total 149)

#### Reconstruction of one-dimensional chaotic maps from sequences of probability density functions

In many practical situations, it is impossible to measure the individual trajectories generated by an unknown chaotic system, but we can observe the evolution of probability density functions generated by such a system. The paper proposes for the first time a matrix-based approach to solve the generalized inverse Frobenius–Perron problem, that is, to reconstruct an unknown ...

#### On the general theory of chaotic dynamics of flexible curvilinear Euler–Bernoulli beams

We present chaotic dynamics of flexible curvilinear shallow Euler–Bernoulli beams. The continuous problem is reduced to the Cauchy problem by the finite-difference method of the second-order accuracy and finite element method (FEM). The Cauchy problem is solved through the fourth- and sixth-order Runge–Kutta methods with respect to time. This preserves reliability of the obtained ...

#### Decomposition frameworks for cooperative manipulation of a planar rigid body with multiple unilateral thrusters

In this paper, we consider cooperative manipulation of a planar rigid body using multiple actuator agents—unilateral thrusters, each attached to the body and each able to apply an unilateral force to the body. Generally, the dynamics of the body manipulated with uncoordinated forces of thrusters is nonlinear. The problem we consider is how to design the unilateral force each agent ...

#### Complex time-delay dynamical systems of quadratic polynomials mapping

Time delays widely exist in the complex dynamical systems. In this paper, time-delay complex dynamical systems are studied by analyzing the properties of corresponding Julia sets. Experimental results show that the stability of complex dynamical system changes greatly due to the influence of time delay. The variations in time-delay Julia sets are studied. In addition, the ...

#### Estimation of the largest Lyapunov exponent-like (LLEL) stability measure parameter from the perturbation vector and its derivative dot product (part 2) experiment simulation

Controlling system dynamics with use of the Largest Lyapunov Exponent (LLE) is employed in many different areas of the scientific research. Thus, there is still need to elaborate fast and simple methods of LLE calculation. This article is the second part of the one presented in Dabrowski (Nonlinear Dyn 67:283–291, 2012). It develops method LLEDP of the LLE estimation and shows that ...

#### Stabilization of the cart pole system: by sliding mode control

This paper presents a control strategy designed as a combination of a PD controller and a twisting-like algorithm to stabilize the damped cart pole system, provided that the pendulum is initially placed within the upper-half plane. To develop the strategy, the original system is transformed into a four-order chain of integrator form, where the damping force is included through an ...

#### Improving the approximation ability of Volterra series identified with a cross-correlation method

This paper proposes an improvement in cross-correlation methods derived from the Lee–Schetzen method, in order to obtain a lower mean square error in the output for a wider range of the input variances. In particular, each Wiener kernel is identified with a different input variance and new formulas for conversion from Wiener to Volterra representation are presented.

#### Conformal invariance and conserved quantity of Mei symmetry for Appell equations in a nonholonomic system of Chetaev’s type

For a nonholonomic system of Chetaev’s type, the conformal invariance and the conserved quantity of Mei symmetry for Appell equations are investigated. First, under the infinitesimal one-parameter transformations of group and the infinitesimal generator vectors, Mei symmetry and conformal invariance of differential equations of motion for the system are defined, and the determining ...

#### On simulation of a bistable system with fractional damping in the presence of stochastic coherence resonance

A bistable dynamical system with the Duffing potential, fractional damping, and random excitation has been modelled. To excite the system, we used a stochastic force defined by Wiener random process of Gaussian distribution. As expected, stochastic resonance appeared for sufficiently high noise intensity. We estimated the critical value of the noise level as a function of ...

#### Periodic solution of a pest management Gompertz model with impulsive state feedback control

In this paper, a new model with two state impulses is proposed for pest management. According to different thresholds, an integrated strategy of pest management is considered, that is to say if the density of the pest population reaches the lower threshold $h_1$ at which pests cause slight damage to the forest, biological control (releasing natural enemy) will be taken to control ...

#### Robust fault-tolerant control for flexible spacecraft against partial actuator failures

In this paper, we present a robust fault-tolerant control scheme to achieve attitude control of flexible spacecraft with disturbances and actuator failures. It is shown that the control algorithms are not only attenuate exogenous bounded disturbances with attenuation level, but also able to tolerate partial loss of actuator effectiveness. The proposed controller design is simple ...

#### Adaptive control of rotary inverted pendulum system with time-varying uncertainties

In this paper, an adaptive controller is proposed to balance a rotary inverted pendulum with time-varying uncertainties. The goal of the control is to bring the pendulum close to the upright position regardless of the various uncertainties and disturbances. Its underactuated dynamics is first decoupled by Olfati’s transformation into a cascade form, and then an adaptive controller ...

#### Resonance solitons produced by azimuthal modulation in self-focusing and self-defocusing materials

We demonstrate azimuthally modulated resonance scalar and vector solitons in self-focusing and self-defocusing materials. They are constructed by selecting appropriately self-consistency and resonance conditions in a coupled system of multicomponent nonlinear Schrödinger equations. In the case with zero modulation depth, it was found that the larger the topological charge, the ...

#### Dynamics of axially accelerating beams with multiple supports

This study represents the transverse vibrations of an axially accelerating Euler–Bernoulli beam resting on multiple simple supports. This is one of the examples of a system experiencing Coriolis acceleration component that renders such systems gyroscopic. A small harmonic variation with a constant mean value for the axial velocity is assumed in the problem. The immovable supports ...

#### Amplitude control approach for chaotic signals

A general approach based on the introduction of a control function for constructing amplitude-controllable chaotic systems with quadratic nonlinearities is discussed in this paper. We consider three control regimes where the control functions are applied to different coefficients of the quadratic terms in a dynamical system. The approach is illustrated using the Lorenz system as a ...