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Nonlinear dynamics near a double Hopf bifurcation for a ship model with time-delay control

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Abstract

A double Hopf bifurcation analysis is performed for the rolling of a low-freeboard ship model controlled with an active U-tube anti-roll tank (ART). We consider a single-degree-of-freedom system with nonlinear damping and restoring functions. The ART is modeled as a proportional-gain controller. A constant delay term is included in the controller since a finite amount of time is required to pump the fluid inside the ART from one container to the other. We perform a linear stability analysis to determine the critical control gain and delay corresponding to the double Hopf bifurcation point. We confirm the existence of the double Hopf bifurcation by finding slopes and roots of the characteristic equation of the linearized delay differential equation with the critical parameters. We use the method of multiple scales to obtain slow-flow equations at the double Hopf bifurcation, which are then used to identify six qualitatively distinct sets of fixed points. Our analysis reveals that one of these regions has a stable zero equilibrium, another has a stable limit cycle with amplitude smaller than the capsizing angle, and the remaining regions have no safe fixed points. These qualitative observations are validated numerically. Study of the control of ship roll motion is important to avoid dynamic instability and capsizing.

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Data are available from the corresponding author on reasonable request.

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Funding

This research received no specific support from any funding source.

Author information

Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, Indian Institute of Technology Hyderabad, Sangareddy, Telangana, 502285, India

    Junaidvali Shaik & C. P. Vyasarayani

  2. Department of Mechanical Engineering, University of Ottawa, 161 Louis-Pasteur, Ottawa, Ontario, K1N 6N5, Canada

    Thomas K. Uchida

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  1. Junaidvali Shaik
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  3. C. P. Vyasarayani
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All authors contributed to the conceptualization and design of the study. Mathematical modeling and numerical simulations were performed by JS. The first draft of the manuscript was written by JS. All authors contributed to the review and editing of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Thomas K. Uchida.

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Appendices

Appendix A: Basis functions used in Eq. (17)

The basis functions used in Eq. (17) are the shifted Legendre polynomials, which are defined as follows:

$$\begin{aligned} \Psi _1(s) =&\ 1, \end{aligned}$$
(A1a)
$$\begin{aligned} \Psi _2(s) =&\ 1 + \frac{2s}{\tau }, \end{aligned}$$
(A1b)
$$\begin{aligned} \Psi _i(s) =&\ \frac{ \left( 2i - 3 \right) \Psi _2(s) \Psi _{i-1}(s) - \left( i-2 \right) \Psi _{i-2}(s)}{i-1},\nonumber \\&\ i = 3, 4, \ldots \ . \end{aligned}$$
(A1c)

When these basis functions are used, the mass matrices \({\varvec{W}}^{(k)}\) and stiffness matrices \({\varvec{Q}}^{(k)}\) in Eq. (22) have the following simple forms:

$$\begin{aligned} W_{ij}^{(k)} =&\ \frac{\tau }{2i-1} \delta _{ij}, \quad i, j = 1, 2, \ldots , N, \end{aligned}$$
(A2)
$$\begin{aligned} Q_{ij}^{(k)} =&\ {\left\{ \begin{array}{ll} 0, &{} \text {if }i \ge j\\ 2, &{} \text {if }i<j\text { and }i+j\text { is odd}\\ 0, &{} \text {if }i<j\text { and }i+j\text { is even} \end{array}\right. },\nonumber \\&\ i,j = 1, 2, \ldots , N. \end{aligned}$$
(A3)

Appendix B: Coefficients \(P_i\) in Eq. (32)

The coefficients \(P_i\) appearing in Eq. (32) have lengthy expressions. Thus, we provide here the expressions for only the pertinent coefficients \(P_1\), \(P_2\), \(P_3\), and \(P_4\).

$$\begin{aligned}&P_1 = \frac{5}{8}\alpha _5A_2^5+\frac{3}{4}\alpha _3 A_2^3+2\omega _{{1}}D_1A_1+2\hat{\mu }_1D_1A_2\nonumber \\&\ +2\Delta k_{cr}\sin (\omega _1\tau _{cr})A_1-2\Delta k_{cr}\cos (\omega _1\tau _{cr})A_2\nonumber \\&\ +\tau _{cr}k_{cr}\sin (\omega _1\tau _{cr})D_1A_1-\tau _{cr}k_{cr}\cos (\omega _1\tau _{cr})D_1A_2\nonumber \\&\ +\frac{3}{4}\mu _3\omega _1^3A_1A_2^2+\frac{15}{4}\alpha _5A_2A_3^2A_4^2+\frac{15}{4}\alpha _5A_1^2A_2A_3^2\nonumber \\&\ +\frac{15}{4}\alpha _5A_1^2A_2A_4^2+\frac{3}{2}\mu _3\omega _1\omega _2^2A_1A_3^2\nonumber \\&\ +\frac{3}{2}\mu _3\omega _1\omega _2^2A_1A_4^2+\frac{15}{4}\alpha _5A_2^3A_3^2+\frac{15}{4}\alpha _5A_2^3A_4^2\nonumber \\&\ +\frac{5}{4}\alpha _5A_1^2A_2^3-\delta \sin (\omega _1\tau _{cr})A_1+\delta \cos (\omega _1\tau _{cr})A_2\nonumber \\&\ +\frac{3}{4}\alpha _3A_1^2A_2+\frac{3}{2}\alpha _3A_2A_3^2+\frac{3}{2}\alpha _3A_2A_4^2\nonumber \\&\ +\frac{3}{4}\mu _3\omega _1^3A_1^3+\frac{15}{8}\alpha _5A_2A_3^4+\frac{5}{8}\alpha _5A_1^4A_2\nonumber \\&\ +\frac{15}{8}\alpha _5A_2A_4^4-2\Delta \omega _0^2A_2-2\Delta \hat{\mu }_1\omega _1A_1, \end{aligned}$$
(B4)
$$\begin{aligned}&P_2 = \frac{3}{4}\alpha _3A_1^3-2\omega _1D_1A_2+2\hat{\mu }_1D_1A_1+\frac{5}{8}\alpha _5A_1^5\nonumber \\&\ +\frac{15}{4}\alpha _5A_1A_3^2A_4^2+\frac{15}{4}\alpha _5A_1A_2^2A_3^2+\frac{15}{4}\alpha _5A_1A_2^2A_4^2\nonumber \\&\ -\tau _{cr}k_{cr}\cos (\omega _1\tau _{cr})D_1A_1-\tau _{cr}k_{cr}\sin (\omega _1\tau _{cr})D_1A_2\nonumber \\&\ -2\Delta k_{cr}\cos (\omega _1\tau _{cr})A_1-2\Delta k_{cr}\sin (\omega _1\tau _{cr})A_2\nonumber \\&\ -\frac{3}{4}\mu _3\omega _1^3A_1^2A_2-\frac{3}{2}\mu _3\omega _1\omega _2^2A_2A_4^2-\frac{3}{2}\mu _3\omega _1\omega _2^2A_2A_3^2\nonumber \\&\ -\frac{3}{4}\mu _3\omega _1^3A_2^3+\frac{15}{8}\alpha _5A_1A_4^4+\frac{5}{8}\alpha _5A_1A_2^4\nonumber \\&\ +\frac{15}{8}\alpha _5A_1A_3^4+\frac{5}{4}\alpha _5A_1^3A_2^2+\frac{15}{4}\alpha _5A_1^3A_3^2\nonumber \\&\ +\frac{15}{4}\alpha _5A_1^3A_4^2+\delta \sin (\omega _1\tau _{cr})A_2+\delta \cos (\omega _1\tau _{cr})A_1\nonumber \\&\ +\frac{3}{2}\alpha _3A_1A_4^2+\frac{3}{4}\alpha _3A_1A_2^2+\frac{3}{2}\alpha _3A_1A_3^2\nonumber \\&\ -2\Delta \omega _0^2 A_1+2\Delta \hat{\mu }_1\omega _1A_2, \end{aligned}$$
(B5)
$$\begin{aligned}&P_3 = \tau _{cr}k_{cr}\sin (\omega _2\tau _{cr})D_1A_3-\tau _{cr}k_{cr}\cos (\omega _2\tau _{cr})D_1A_4\nonumber \\&\ +2\Delta k_{cr}\sin (\omega _2\tau _{cr})A_3-2\Delta k_{cr}\cos (\omega _2\tau _{cr})A_4\nonumber \\&\ +\frac{3}{4}\mu _3\omega _2^3A_3A_4^2+\frac{15}{4}\alpha _5A_1^2A_2^2A_4+\frac{15}{4}\alpha _5A_1^2A_3^2A_4\nonumber \\&\ +\frac{15}{4}\alpha _5A_2^2A_3^2A_4+\frac{15}{4}\alpha _5A_2^2A_4^3+\frac{15}{4}\alpha _5A_1^2A_4^3\nonumber \\&\ +\frac{5}{4}\alpha _5A_3^2A_4^3-\delta \sin (\omega _2\tau _{cr})A_3+\delta \cos (\omega _2\tau _{cr})A_4\nonumber \\&\ +\frac{3}{2}\alpha _3A_1^2A_4+\frac{3}{2}\alpha _3A_2^2A_4+\frac{3}{4}\alpha _3A_3^2A_4\nonumber \\&\ +\frac{3}{4}\mu _3\omega _2^3A_3^3+\frac{15}{8}\alpha _5A_2^4A_4+\frac{15}{8}\alpha _5A_1^4A_4\nonumber \\&\ +\frac{5}{8}\alpha _5A_3^4A_4+\frac{5}{8}\alpha _5A_4^5+\frac{3}{4}\alpha _3A_4^3+2\omega _2D_1A_3\nonumber \\&\ +2\hat{\mu }_1D_1A_4+\frac{3}{2}\mu _3\omega _1^2\omega _2A_1^2A_3+\frac{3}{2}\mu _3\omega _1^2\omega _2A_2^2A_3\nonumber \\&\ -2\Delta \hat{\mu }_1\omega _2A_3-2\Delta \omega _0^2A_4, \end{aligned}$$
(B6)
$$\begin{aligned}&P_4 = \frac{5}{8}\alpha _5A_3^5+\frac{3}{4}\alpha _3A_3^3-2\omega _2D_1A_4+2\hat{\mu }_1D_1A_3\nonumber \\&\ +\frac{15}{4}\alpha _5A_1^2A_2^2A_3+\frac{15}{4}\alpha _5A_1^2A_3A_4^2+\frac{15}{4}\alpha _5A_2^2A_3A_4^2\nonumber \\&\ -2\Delta \omega _0^2A_3+2\Delta \omega _2\hat{\mu }_1A_4-\tau _{cr}k_{cr}\cos (\omega _2\tau _{cr})D_1A_3\nonumber \\&\ -\tau _{cr}k_{cr}\sin (\omega _2\tau _{cr})D_1A_4+\frac{5}{4}\alpha _5A_3^3A_4^2\nonumber \\&\ -\frac{3}{4}\mu _3\omega _2^3A_3^2A_4-\frac{3}{2}\mu _3\omega _1^2\omega _2A_1^2A_4-\frac{3}{2}\mu _3\omega _1^2\omega _2A_2^2A_4\nonumber \\&\ +\frac{15}{4}\alpha _5A_2^2A_3^3+\frac{15}{4}\alpha _5A_1^2A_3^3+\frac{3}{2}\alpha _3A_1^2A_3\nonumber \\&\ +\frac{3}{2}\alpha _3A_2^2A_3+\frac{3}{4}\alpha _3A_3A_4^2-\frac{3}{4}\mu _3\omega _2^3A_4^3\nonumber \\&\ +\frac{15}{8}\alpha _5A_2^4A_3+\frac{15}{8}\alpha _5A_1^4A_3+\frac{5}{8}\alpha _5A_3A_4^4\nonumber \\&\ +\delta \sin (\omega _2\tau _{cr})A_4+\delta \cos (\omega _2\tau _{cr})A_3\nonumber \\&\ -2\Delta k_{cr}A_3\cos (\omega _2\tau _{cr})-2\Delta k_{cr}A_4\sin (\omega _2\tau _{cr}), \end{aligned}$$
(B7)

where \(A_i = A_i(T_1)\) and \(\displaystyle {D_1 A_i = \frac{d A_i(T_1)}{d T_1}}\).

Appendix C: Coefficients \(N_i\) in Eqs. (36)

The expressions for the coefficients in Eqs. (36) are as follows:

$$\begin{aligned} H_1= & {} 8 k_{cr} \tau _{cr} \big ( 4 \omega _1 \sin (\omega _1 \tau _{cr}) - 4 \hat{\mu }_1 \cos (\omega _1 \tau _{cr})\\{} & {} + k_{cr} \tau _{cr} \big ) + 32 \left( \hat{\mu }_1^2 + \omega _1^2 \right) ,\\ H_2= & {} 8 k_{cr} \tau _{cr} \big ( 4 \omega _2 \sin (\omega _2 \tau _{cr}) - 4 \hat{\mu }_1 \cos (\omega _2 \tau _{cr})\\{} & {} + k_{cr} \tau _{cr} \big ) + 32 \left( \hat{\mu }_1^2 + \omega _2^2 \right) ,\\ N_{11}= & {} 5 \hat{\alpha }_5 \left( k_{cr} \tau _{cr} \cos (\omega _1 \tau _{cr}) - 2 \hat{\mu }_1 \right) ,\\ N_{12}= & {} 15 \hat{\alpha }_5 \left( k_{cr} \tau _{cr} \cos (\omega _1 \tau _{cr}) - 2 \hat{\mu }_1 \right) ,\\ N_{13}= & {} 30 \hat{\alpha _5} \left( k_{cr} \tau _{cr} \cos (\omega _1 \tau _{cr}) - 2 \hat{\mu }_1 \right) ,\\ N_{14}= & {} 6 \big [ \hat{\alpha }_3 \left( k_{cr} \tau _{cr} \cos (\omega _1 \tau _{cr}) - 2 \hat{\mu }_1 \right) \\{} & {} - \hat{\mu }_3 \omega _1^3 \left( k_{cr} \tau _{cr} \sin (\omega _1 \tau _{cr}) + 2 \omega _1 \right) \big ],\\ N_{15}= & {} 12 \big [ \hat{\alpha }_3 \left( k_{cr} \tau _{cr} \cos (\omega _1 \tau _{cr}) - 2 \hat{\mu }_1 \right) \\{} & {} - \hat{\mu }_3 \omega _1 \omega _2^2 \left( k_{cr} \tau _{cr} \sin (\omega _1 \tau _{cr}) - 2 \omega _1 \right) \big ],\\ N_{16}= & {} 16 \big [ 2 \hat{\mu }_1 \left( \omega _0^2 + \omega _1^2 \right) + k_{cr} \omega _1 \left( \tau _{cr} \hat{\mu }_1 - 2 \right) \\{} & {} \sin (\omega _1 \tau _{cr}) - k_{cr} \left( \tau _{cr} \omega _0^2 - 2 \hat{\mu }_1 \right) \cos (\omega _1 \tau _{cr})\\{} & {} - k_{cr}^2 \tau _{cr} \big ],\\ N_{17}= & {} 16 \left( \omega _1 \sin (\omega _1 \tau _{cr}) - \hat{\mu }_1 \cos (\omega _1 \tau _{cr}) \right) + 8 k_{cr} \tau _{cr},\\ N_{21}= & {} 15 \hat{\alpha }_5 \left( k_{cr} \tau _{cr} \cos (\omega _2 \tau _{cr}) - 2 \hat{\mu }_1 \right) ,\\ N_{22}= & {} 5 \hat{\alpha }_5 \left( k_{cr} \tau _{cr} \cos ( \omega _2 \tau _{cr}) - 2 \hat{\mu }_1 \right) ,\\ N_{23}= & {} 30 \hat{\alpha }_5 \left( k_{cr} \tau _{cr} \cos ( \omega _2 \tau _{cr}) - 2 \hat{\mu }_1 \right) ,\\ N_{24}= & {} 12 \big [ \hat{\alpha }_3 \left( k_{cr} \tau _{cr} \cos (\omega _2 \tau _{cr}) - 2 \hat{\mu }_1 \right) \\{} & {} - \hat{\mu }_3 \omega _1^2 \omega _2 \left( k_{cr} \tau _{cr} \sin (\omega _2 \tau _{cr}) + 2 \omega _2 \right) \big ],\\ N_{25}= & {} 6 \big [ \hat{\alpha }_3 \left( k_{cr} \tau _{cr} \cos (\omega _2 \tau _{cr}) - 2 \hat{\mu }_1 \right) \\{} & {} - \hat{\mu }_3\omega _2^3 \left( k_{cr} \tau _{cr} \sin (\omega _2 \tau _{cr}) + 2 \omega _2 \right) \big ],\\ N_{26}= & {} 16 \big [ 2 \hat{\mu }_1 \left( \omega _0^2 + \omega _2^2 \right) + k_{cr} \omega _2 \left( \tau _{cr} \hat{\mu }_1 - 2 \right) \\{} & {} \sin (\omega _2 \tau _{cr}) - k_{cr} \left( \tau _{cr} \omega _0^2 - 2 \hat{\mu }_1 \right) \cos (\omega _2 \tau _{cr})\\{} & {} - k_{cr}^2 \tau _{cr} \big ],\\ N_{27}= & {} 16 \left( \omega _2 \sin (\omega _2 \tau _{cr}) - \hat{\mu }_1 \cos (\omega _2 \tau _{cr}) \right) + 8 k_{cr} \tau _{cr},\\ N_{31}= & {} 5 \hat{\alpha }_5 \left( k_{cr} \tau _{cr} \sin (\omega _1 \tau _{cr}) + 2 \omega _1 \right) ,\\ N_{32}= & {} 15 \hat{\alpha }_5 \left( k_{cr} \tau _{cr} \sin (\omega _1 \tau _{cr}) + 2 \omega _1 \right) ,\\ N_{33}= & {} 30 \hat{\alpha }_5 \left( k_{cr} \tau _{cr} \sin (\omega _1 \tau _{cr}) + 2 \omega _1 \right) ,\\ N_{34}= & {} 6 \big [ \hat{\mu }_3 \omega _1^3 \left( k_{cr} \tau _{cr} \cos (\omega _1 \tau _{cr}) - 2 \hat{\mu }_1 \right) \\{} & {} + \hat{\alpha }_3 \left( k_{cr} \tau _{cr} \sin (\omega _1 \tau _{cr}) + 2 \omega _1 \right) \big ],\\ N_{35}= & {} 12 \big [ \hat{\mu }_3 \omega _1 \omega _2^2 \left( k_{cr} \tau _{cr} \cos (\omega _1 \tau _{cr}) - 2 \hat{\mu }_1 \right) \\{} & {} + \hat{\alpha }_3 \left( k_{cr} \tau _{cr} \sin (\omega _1 \tau _{cr}) + 2 \omega _1 \right) \big ],\\ N_{36}= & {} 16 \big [ \omega _1 \big ( 2 \hat{\mu }_1^2 - 2 \omega _0^2 - k_{cr} \left( 2 + \tau _{cr} \hat{\mu }_1 \right) \\{} & {} \cos (\omega _1 \tau _{cr}) \big ) - k_{cr} \left( 2 \hat{\mu }_1 + \tau _{cr} \omega _0^2 \right) \sin (\omega _1 \tau _{cr}) \big ],\\ N_{37}= & {} 16 \left( \hat{\mu }_1 \sin (\omega _1 \tau _{cr}) + \omega _1 \cos (\omega _1 \tau _{cr}) \right) ,\\ \end{aligned}$$
$$\begin{aligned} N_{41}= & {} 15 \hat{\alpha }_5 \left( k_{cr} \tau _{cr} \sin (\omega _2 \tau _{cr}) + 2 \omega _2 \right) ,\\ N_{42}= & {} 5 \hat{\alpha }_5 \left( k_{cr} \tau _{cr} \sin (\omega _2 \tau _{cr}) + 2 \omega _2 \right) ,\\ N_{43}= & {} 30 \hat{\alpha }_5 \left( k_{cr} \tau _{cr} \sin (\omega _2 \tau _{cr}) + 2 \omega _2 \right) ,\\ N_{44}= & {} 12 \big [ \hat{\mu }_3 \omega _2 \omega _1^2 \left( k_{cr} \tau _{cr} \cos (\omega _2 \tau _{cr}) - 2 \hat{\mu }_1 \right) \\{} & {} + \hat{\alpha }_3 \left( k_{cr} \tau _{cr} \sin (\omega _2 \tau _{cr}) + 2 \omega _2 \right) \big ],\\ N_{45}= & {} 6 \big [ \hat{\mu }_3 \omega _2^3 \left( k_{cr} \tau _{cr} \cos (\omega _2 \tau _{cr}) - 2 \hat{\mu }_1 \right) \\{} & {} + \hat{\alpha }_3 \left( k_{cr} \tau _{cr} \sin (\omega _2 \tau _{cr}) + 2 \omega _2 \right) \big ],\\ N_{46}= & {} 16 \big [ \omega _2 \big ( 2 \hat{\mu }_1^2 - 2 \omega _0^2 - k_{cr} \left( 2 + \tau _{cr} \hat{\mu }_1 \right) \\{} & {} \cos (\omega _2 \tau _{cr}) \big ) - k_{cr} \left( 2 \hat{\mu }_1 + \tau _{cr} \omega _0^2 \right) \sin (\omega _2 \tau _{cr}) \big ],\\ N_{47}= & {} 16 \left( \hat{\mu }_1 \sin (\omega _2 \tau _{cr}) + \omega _2 \cos (\omega _2 \tau _{cr}) \right) . \end{aligned}$$

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Shaik, J., Uchida, T.K. & Vyasarayani, C.P. Nonlinear dynamics near a double Hopf bifurcation for a ship model with time-delay control. Nonlinear Dyn 111, 21441–21460 (2023). https://doi.org/10.1007/s11071-023-08965-y

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