Periodic behavior of a class of nonlinear dynamic systems based on the Runge-Kutta algorithm

Abstract

In this paper, the periodic state of a class of nonlinear hydrodynamic models is studied. This study is applicable to many fields, such as engineering, cybernetics and so on. We use the Runge Kutta algorithm. First, we give a four order nonlinear hydrodynamic model based on the Runge Kutta algorithm. Then, we study the periodic dynamic characteristics of the nonlinear model and establish some new sufficient conditions. The results of our study promote and improve the existing results. And the numerical results are also verified by the numerical experiments. Therefore, the study of this paper has a great help to master the periodic motion of the fluid.

Keywords

Runge-Kutta algorithm nonlinear dynamic system periodic solution Liapunov function method

1 Introduction

The Runge-Kutta algorithm we know well is a kind of high precision single step algorithm which is widely used. One of the various Runge-Kutta algorithms is so important that it is often referred to as the “RK4” method [1]. This method is mainly based on the information of the derivative and initial value of the equation, and uses the computer simulation to save the complex process of solving differential equations [2]. It is difficult to see the general characteristics of some specific periodic orbits by using the numerical method to study the periodicity [3]. Now, by using the Runge-Kutta algorithm and the “RK4” method, combined with the Liapunov function method to study the periodicity, we can get the effective result of periodic solutions[4, 5].

Fluid flow problems are widely used in many engineering fields, such as: the flow of oil in the pipeline, with processing flow in the mould body of polymer materials, changes in the muddy seabed tidal current and wave force factors and the characteristics of sediment with viscoelastic fluid, and movement with the tide and wave occurrence. The discussion of a large number of practical problems can be attributed to nonlinear differential equations [6 –8]. By using the method of fixed point analysis of functional analysis, the generalized asymptotic solutions of singular perturbation problems for two parameter nonlocal generalized parabolic equations are constructed by stretching variable transformation and boundary layer and initial layer method [9]. Differential equations can be used to describe some laws of fluid motion. In a nonlinear fluid dynamic system, periodic motion is the most important. Since tidal currents and waves are periodic motions, it is very important for us to understand the motion rules of fluids under external forces. Based on the classical harmonic balance method a new technique is presented to determine periodic solutions of the nonlinear differential equations. But this method is not applicable to the four order differential equation [10 –12]. The nonlinear periodic differential system is processed by small parameter. In the small parameter, the expression of the two order bifurcation function is given, which makes the simple zero is the initial value of the periodic solution which still exists after disturbance. The periodic solution is obtained by using the Liapunov Schmidt reduction method [13]. With the progress of computer technology and the improvement of fluid mechanics theory, people usually rely on numerical methods for solving higher order differential dynamical systems, for example, including lindsted Poincare method, averaging method and multi-scale method [14 –16]. But there is a disadvantage of this method: it has great error to the nonlinear system [17]. Therefore, it is very important to study the periodic solution of the nonlinear dynamic system.

In this paper, a class of nonlinear fluid dynamics systems is studied. By using the Runge-Kutta algorithm and the “RK4” method and the Liapunov function method, we get the existence and uniqueness of the periodic solution of the power system. It helps us to understand the periodic motion of fluid flow.

2 Model and methodology

Because of the importance of fluid flow problem, its rule can be expressed by differential equation. Therefore, the fluid flow problem described in this paper can be expressed by a nonlinear differential equation model: $x^{(4)} + P (t, \overset{⃛}{x}) + Q (t, \ddot{x}) + H (t, \dot{x}) + Φ (t, x) = 0$ (1)

The system (1) is very important in both theory and application. The periodic solution of the dynamical system (1) is obtained by using the Runge-Kutta algorithm and the “RK4” method and combining the Liapunov function method. It is of great significance to understand and understand the periodic motion of the fluid flow.

3 Results and discussion

In this paper, in order to study the periodic solution of the system (1). We consider the system (1) or its equivalent. ${\begin{matrix} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = x_{3} \\ {\dot{x}}_{3} = x_{4} \\ {\dot{x}}_{4} = - d (t) x_{1} - c (t) x_{2} - b (t) x_{3} - a (t) x_{4} - φ (t, x_{1}) \\ - h (t, x_{2}) - q (t, x_{3}) - p (t, x_{4}) \end{matrix}$ (2) where $P (t, \overset{⃛}{x}) = a (t) \overset{⃛}{x} + p (t, \overset{⃛}{x}), Q (t, \ddot{x}) = b (t) \ddot{x} + q (t, \ddot{x}),$ $H (t, \dot{x}) = c (t) \dot{x} + h (t, \dot{x}), Φ (t, x) = d (t) x + φ (t, x) .$

These functions are continuous and differentiable with respect to each variable and are periodic with period to t with same period ω. The coefficient matrix for (2). $A (t) = [\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ - d (t) & - c (t) & - b (t) & - a (t) \end{matrix}]$ (3)

Theorem 1. Suppose system (2) satisfies the following conditions:

All the characteristic roots λ_i (t) of the generalized characteristic equation of the coefficient matrix (3) have negative real parts, $Re λ_{i} (t) \leq - δ < 0, (i = 1, 2, 3, 4) .$ where δ is a positive constant;

a (t) , b (t) , c (t) , d (t) are all positive bounded with the same upper bound M on [0, + ∞], M>1;

$| \dot{a} (t) | \leq η$ , $| \dot{b} (t) | \leq η$ , $| \dot{c} (t) | \leq η$ , $| \dot{d} (t) | \leq η$ , t ∈ [0, + ∞], $0 < η \leq \frac{25 δ^{10}}{32 M^{3}}$

$lim_{ρ \to + \infty} \frac{| p (t, x_{4}) |}{ρ} = μ_{1}$ , $lim_{ρ \to + \infty} \frac{| q (t, x_{3}) |}{ρ} = μ_{2}$ , $lim_{ρ \to + \infty} \frac{| h (t, x_{2}) |}{ρ} = μ_{3}$ , $lim_{ρ \to + \infty} \frac{| φ (t, x_{1}) |}{ρ} = μ_{4}$ $ρ = {(x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2})}^{1 / 2},$ $μ = max {μ_{1}, μ_{2}, μ_{3}, μ_{4}}, 0 < μ \leq \frac{δ^{10}}{7 M^{4}} .$

Then system (2) has at least one periodic solution with period ω.

Proof. The generalized characteristic equation for the coefficient matrix (3) is $λ^{4} + a λ^{3} + b λ^{2} + c λ + d = 0 .$

Then we have $\begin{matrix} a > 0, b > 0, c > 0, d > 0, ab - c > 0, \\ abc - c^{2} - a^{2} d > 0 . \\ a = - (λ_{1} + λ_{2} + λ_{3} + λ_{4}) \geq 4 δ . \\ b = λ_{1} λ_{2} + λ_{1} λ_{3} + λ_{1} λ_{4} + λ_{2} λ_{3} + λ_{2} λ_{4} + λ_{3} λ_{4} \geq 6 δ^{2} . \\ c = - (λ_{1} λ_{2} λ_{3} + λ_{1} λ_{2} λ_{4} + λ_{1} λ_{3} λ_{4} + λ_{2} λ_{3} λ_{4}) \geq 4 δ^{3} . \\ d = λ_{1} λ_{2} λ_{3} λ_{4} \geq δ^{4} . ab - c \geq 20 δ^{3} \\ abc - c^{2} - a^{2} d \geq 64 δ^{6} . \end{matrix}$ where all the characteristic roots λ_i (t) of the generalized characteristic equation of the coefficient matrix (3) have negative real parts, $Re λ_{i} (t) \leq - δ < 0, (i = 1, 2, 3, 4)$

We construct Liapunov function $V = V_{1} + V_{2} + V_{3} + V_{4} .$ (4) where $\begin{matrix} V_{1} = \frac{1}{2 c} (abc - c^{2} - a^{2} d) ({cx}_{1} + {bx}_{2} + {ax}_{3} + x_{4})^{2} + \\ \frac{d}{2 c} [(ab - c) x_{2} + a^{2} x_{3} + {ax}_{4}]^{2} + \\ \frac{d}{2 a} [(ab - c) x_{1} + a^{2} x_{2} + {ax}_{3}]^{2} + \\ \frac{d}{2 a} (abc - a^{2} d - c^{2}) x_{1}^{2} . \\ V_{2} = \frac{ad}{2} {(x_{4} + {ax}_{3} + \frac{ab - c}{a} x_{2})}^{2} + \\ \frac{cd}{2} {(x_{3} + {ax}_{2} + \frac{ad}{c} x_{1})}^{2} + \\ \frac{d^{2}}{2 c} (abc - c^{2} - a^{2} d) x_{1}^{2} + \frac{d}{2 a} (abc - c^{2} - a^{2} d) x_{2}^{2} \\ V_{3} = \frac{ad}{2} {({dx}_{1} + {cx}_{2} + \frac{c}{a} x_{3})}^{2} + \\ \frac{cd}{2} {(x_{4} + {ax}_{3} + \frac{ad}{c} x_{2})}^{2} + \\ \frac{d^{2}}{2 c} (abc - c^{2} - a^{2} d) x_{2}^{2} + \frac{d}{2 a} (abc - c^{2} - a^{2} d) x_{3}^{2} . \\ V_{4} = \frac{cd}{2} {({dx}_{1} + {cx}_{2} + \frac{bc - ad}{c} x_{3})}^{2} + \\ \frac{ad}{2} {({dx}_{2} + {cx}_{3} + \frac{c}{a} x_{4})}^{2} + \\ \frac{d^{2}}{2 c} (abc - c^{2} - a^{2} d) x_{3}^{2} + \frac{d}{2 a} (abc - c^{2} - a^{2} d) x_{4}^{2} \\ V \geq V_{2} + V_{4} \geq d (abc - c^{2} - a^{2} d) [\frac{d}{2 c} (x_{1}^{2} + x_{3}^{2}) + \\ \frac{1}{2 a} (x_{2}^{2} + x_{4}^{2})] \\ \geq δ^{4} \cdot 64 δ^{6} [\frac{δ^{4}}{2 M} (x_{1}^{2} + x_{3}^{2}) + \frac{1}{2 M} (x_{2}^{2} + x_{4}^{2})] \\ \geq \frac{32 B δ^{10}}{M} (x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2}) \end{matrix}$ where B = min(δ⁴, 1), hence V is a position definite and we get $V \leq 17 M^{7} (x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2})$ hence V has an infinitesimal upper bound.□

Taking the total derivative of V with respect to t along the trajectory of (2), we have $\begin{matrix} {\frac{dV}{dt} |}_{(2)} = \frac{\partial V}{\partial t} + \frac{\partial V}{\partial x_{1}} x_{2} + \frac{\partial V}{\partial x_{2}} x_{3} + \frac{\partial V}{\partial x_{3}} x_{4} + \frac{\partial V}{\partial x_{4}} \\ [- d (t) x_{1} - c (t) x_{2} - b (t) x_{3} - a (t) x_{4} - φ (t, x_{1}) \\ - h (t, x_{2}) - q (t, x_{3}) - p (t, x_{4})] \end{matrix}$ $\leq | \frac{\partial V}{\partial t} | - d (abc - c^{2} - a^{2} d) (x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2})$ $+ \frac{\partial V}{\partial x_{4}} [- φ (t, x_{1}) - h (t, x_{2}) - q (t . x_{3}) - p (t, x_{4})$ $| \frac{\partial V}{\partial t} | \leq 64 η M^{3} (x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2}) .$ $| \frac{\partial V}{\partial x_{4}} | \leq 14 M^{4} \sqrt{x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2}} .$

We conclude that for an arbitrarily given $ɛ_{0} (0 < ɛ_{0} < \frac{3 δ^{10}}{70 M^{4}})$ , there exists a sufficiently large number N > 0 such that $x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} \geq N^{2}$ will lead to $| p (t, x_{4}) | \leq (μ + ɛ_{0}) ρ, | q (t, x_{3}) | \leq (μ + ɛ_{0}) ρ,$ $| h (t, x_{2}) | \leq (μ + ɛ_{0}) ρ, | φ (t, x_{1}) | \leq (μ + ɛ_{0}) ρ .$

Therefore, in the product space $\begin{matrix} (0 \leq t < + \infty) \times {(x_{1}, x_{2}, x_{3}, x_{4}) | x_{1}^{2} + x_{2}^{2} \\ + x_{3}^{2} + x_{4}^{2} \geq N^{2}} \end{matrix}$ we have $\begin{matrix} {\frac{dV}{dt} |}_{(3)} \leq [64 η M^{3} - 64 δ^{10} + 70 (μ + ɛ_{0}) M^{4}] \\ (x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2}) \end{matrix}$ $= - [δ^{10} + (50 δ^{10} - 64 η M^{3}) + (10 δ^{10} - 70 μ M^{4})$ $+ (3 δ^{10} - 70 ɛ_{0} M^{4})] (x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2})$ $\leq - δ^{10} (x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2})$

The nonlinear system (2) has at least one periodic solution with period ω.

Theorem 2. Suppose system (2) satisfies the following conditions:

The conditions (i),(ii),(iii) of Theorem l;

All the function p, q, h, φ in system (2) have continuous first-order partial derivatives with respect to each variable. There exists a positive constant $\begin{matrix} lim_{x_{4}^{2} (s) \to \infty} | \int_{0}^{1} p_{x_{4}}^{'} (t, x_{4}) ds | < β, \\ lim_{x_{1}^{2} (s) \to \infty} | \int_{0}^{1} φ_{x_{1}}^{'} (t, x_{1}) ds | < β, \\ lim_{x_{3}^{2} (s) \to \infty} | \int_{0}^{1} q_{x_{i}}^{'} (t, x_{3}) ds | < β, \\ lim_{x_{2}^{2} (s) \to \infty} | \int_{0}^{1} h_{x_{j}}^{'} (t, x_{2}) ds | < β . \end{matrix}$

Then system (2) has a unique asymptotically stable periodic solution with period ω.

Proof. Let (x₁, x₂, x₃, x₄) and (z₁, z₂, z₃, z₄) are two arbitrary solution of system (3), then they should satisfy $\frac{d (x_{1} - z_{1})}{dt} = x_{2} - z_{2}$ $\frac{d (x_{2} - z_{2})}{dt} = x_{3} - z_{3}$ $\frac{d (x_{3} - z_{3})}{dt} = x_{4} - z_{4},$ $\frac{d (x_{4} - z_{4})}{dt} = - d (x_{1} - z_{1}) - c (x_{2} - z_{2}) - b (x_{3} - z_{3})$ $- a (x_{4} - z_{4}) - [φ (t, x_{1}) - φ (t, z_{1})$ $- [h (t, x_{2}) - h (t, z_{2})] - [q (t, x_{3}) - q (t, z_{3})]$ $- [p (t, x_{4}) - p (t, z_{4})]$

According to integral mean value theorem, we have $\begin{matrix} φ (t, x_{1}) - φ (t, z_{1}) = (x_{1} - z_{1}) \\ \int_{0}^{1} φ^{'} (t, {sx}_{1} + (1 - s) z_{1}) ds \underline{\underline{Δ}} (x_{1} - z_{1}) \\ \int_{0}^{1} φ^{'} ds \end{matrix}$

Similarly, we obtain $h (t, x_{2}) - h (t, z_{2}) = (x_{2} - z_{2}) \int_{0}^{1} h^{'} ds .$

$p (t, x_{4}) - p (t, z_{4}) = (x_{4} - z_{4}) \int_{0}^{1} p^{'} ds .$

$q (t, x_{3}) - q (t, z_{3}) = (x_{3} - z_{3}) \int_{0}^{1} q^{'} ds .$

Let u_i = x_i - z_i (i = 1, 2, 3, 4) , we have

${\begin{matrix} \frac{{du}_{1}}{dt} = u_{2} \\ \frac{{du}_{2}}{dt} = u_{3} \\ \frac{{du}_{3}}{dt} = u_{4} \\ \frac{{du}_{4}}{dt} = - {du}_{1} - {cu}_{2} - {bu}_{3} - {au}_{4} \\ - u_{1} \int_{0}^{1} φ^{'} ds - u_{2} \int_{0}^{1} h^{'} ds - u_{3} \int_{0}^{1} q^{'} ds - u_{4} \int_{0}^{1} p^{'} ds \end{matrix}$ (5)

From condition 2), for an arbitrarily given $ɛ_{1} (0 < ɛ_{1} < \frac{3 δ^{10}}{148 M^{4}})$ , there must exists a sufficiently large number N₁ (N₁ > N), such that $x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} \geq N_{1}^{2}$ will lead to $| \int_{0}^{1} p^{'} ds | < β + ɛ_{1}, | \int_{0}^{1} φ^{'} ds | < β + ɛ_{1},$ $| \int_{0}^{1} q^{'} ds | < β + ɛ_{1}, | \int_{0}^{1} h^{'} ds | < β + ɛ_{1} .$

Now we still take (4) as the Liapunov function of system (5), only with x_i replaced u_i (i = 1, 2, 3, 4). Thus we can obtain the total derivative of V with respect to t along the trajectory of (5). ${\frac{dV}{dt} |}_{(5)} = \frac{\partial V}{\partial t} + \frac{\partial V}{\partial u_{1}} u_{2} + \frac{\partial V}{\partial u_{2}} u_{3} + \frac{\partial V}{\partial u_{3}} u_{4}$ $+ \frac{\partial V}{\partial u_{4}} [- d (t) u_{1} - c (t) u_{2} - b (t) u_{3} - a (t) u_{4}$ $\begin{matrix} - u_{1} \int_{0}^{1} φ^{'} ds - u_{2} \int_{0}^{1} h^{'} ds - u_{3} \int_{0}^{1} q^{'} ds - u_{4} \\ \int_{0}^{1} p_{4}^{'} ds \leq [64 η M^{3} - 64 δ^{10} + 148 (β + ɛ_{1}) M^{4}] \\ (u_{1}^{2} + u_{2}^{2} + u_{3}^{2} + u_{4}^{2}) \leq - δ^{10} (u_{1}^{2} + u_{2}^{2} + u_{3}^{2} + u_{4}^{2}) \end{matrix}$

It is obvious that in the product space $\begin{matrix} I (0 \leq t < + \infty) \times {(u_{1}, u_{2}, u_{3}, u_{4}) | u_{1}^{2} + u_{2}^{2} \\ + u_{3}^{2} + u_{4}^{2} \geq N_{1}^{2}} \end{matrix}$

${\frac{dV}{dt} |}_{(5)}$ is negatively definite. Thereby the zero solution of system (5) is asymptotically stable. Thus the original system (2) is extremely stable. Therefore, according to Lemma 3, system (2) has a unique asymptotically stable periodic solution with period ω.□

4 Conclusion

In this paper, the Runge-Kutta algorithm and the “RK4” method are used to study a class of nonlinear dynamical systems. The periodic solutions and characteristics of the system (1) are obtained. The periodic motion of the fluid is revealed. The results show that the fluid movement changes periodically. In this paper, the fluid motion is periodic and stable. It can provide a scientific basis for the design of efficient pipeline transportation technology, and is of great significance for us to understand the periodic phenomena of fluid motion.

Footnotes

Acknowledgments

This work was supported by Chinese Natural Science Foundation (No.11361048), Yunnan Natural Science Foundation (No. 2017FH001-014), Yunnan Science Foundation (No. 2019J0613) and Qujing Normal University Science Foundation (No. ZDKC2016002).

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