Effects of passive hydrodynamics force on harmonic and chaotic oscillations in nonlinear chemical dynamics

Nonlinear Dynamics of the Quadratic-Damping Helmholtz Oscillator

“…In this part, we use the harmonic balance method to study harmonic oscillations. For this, we take the solutions of equation ( 8) in the following form [18,19,37]:…”

“…In order to analyze the primary and secondary resonances, we use the multiple time-scale method; the solution of equation ( 8) can be expressed in the following form [18][19][20]: Complexity where T 0 � τ, T 1 � ετ, and ε is the small pertubative parameter. e time derivatives transform into…”

“…e search for these phenomena for the new Helmholtz model therefore seems very important. e determination of the states of resonance in a nonlinear dynamic system makes it possible to predict energy exchanges by making the energy proportional to the square of the amplitude of the vibrations [18][19][20][21]. For example, in mechanical systems, a sudden increase in the resonance energy can cause damage to the mechanical system, while antiresonance systems can be used to store energy.…”

“…us, the condition giving the relation between the amplitude of vibration and the parameters of the system is crucial for the good choice of the frequency and the amplitude of excitation to trigger a nonlinear resonance effect [34][35][36]. e most suitable techniques for finding resonances are the harmonic balance and the multiple-scale methods [18,19,37]. e object of chaos theory is the study of nonlinear phenomena governed by simple and deterministic laws whose behavior under certain conditions becomes unpredictable.…”

“…e object of chaos theory is the study of nonlinear phenomena governed by simple and deterministic laws whose behavior under certain conditions becomes unpredictable. Since its discovery in the 20th century, chaos has been one of the most interesting for dynamic systems in areas such as physics, mathematics, chemistry, biochemistry, economics and finance, epidemiology, and engineering [19][20][21][22][23][24][25]. Depending on the field of study, it is sometimes useful or undesirable to the point where many researchers are interested in its prediction and/or control.…”