1 Introduction

In 1927, the renowned Dutch electrical engineer Balthazar van der Pol first derived what is now known as the famous van der Pol equation to describe the oscillatory effect of auditions in electronic circuits. Since then, scholars have published numerous papers utilizing this equation to investigate nonlinear dynamical systems and seek improved solutions. The van der Pol oscillation system, regarded as a classical self-excited oscillation system, has become a significant mathematical model extensively applied in the modeling of more intricate dynamic systems in physics, radio electronics, building science, automation technology, and even medicine. The van der Pol oscillation has evolved into a fundamental model for describing the process of oscillation [1,2,3,4,5,6,7,8,9]. The nonlinear component of a van der Pol-Duffing system encompasses both the third-order nonlinear restoring force term from the Duffing system and the nonlinear damping term from the van der Pol system to sustain self-excited oscillations. For decades, scholars have systematically and comprehensively investigated the dynamic behavior of van der Pol-Duffing systems. El-Dib Yusry O presented an effective method for solving fractional Van der Pol-Duffing oscillators [10]. Vinicius Wiggers et al. examined multi-stability and periodic organization within Van der Pol-Duffing oscillators [11]. Abdelnebi Amira et al. studied existence, uniqueness, and stability of a novel class of sequential Van der Pol-Duffing fractional differential oscillator solutions with acceleration using Caputo-Hadamard derivatives [12].

With a history spanning over 300 years, the concept of nearly classical calculus has garnered favor among numerous scholars since its inception. As research on fractional calculus theory and application deepens, it has experienced rapid development since 1974, encompassing an increasing number of fields and extending to the popularization of fractional theories in ordinary differential equations and even functional differential equations. Simultaneously, fractional calculus serves as the foundation and invaluable tool for fractal geometry and fractal dimension dynamics. Significant advancements have been achieved across various domains including oscillation, control systems, diffusion and transport theory, biological organization, organic chemical industry, and medicine [13,14,15,16,17,18,19,20]. The use of fractional derivative models addresses a critical limitation where classical integer differential model theories fail to align well with experimental results; instead yielding relatively better outcomes with fewer parameters. When describing complex physics and mechanics problems, the physical interpretation of fractional derivative models is clearer and more concise compared to nonlinear models. For instance, fractional control is actively introduced into vibration systems to achieve superior vibration control effects [21]. Viscoelastic devices within vibration systems are accurately modeled and analyzed using fractional calculus [22].

In the van der Poel-Dufenfing system, the introduction of fractional calculus can be used to describe memory effects and long-range correlations in the system. Physically, fractional calculus can be interpreted as a non-instantaneous response of a system to an external stimulus. Traditional integer calculus describes the system's response to instantaneous excitations, while fractional calculus considers the long-term cumulative effect of the system on the excitation, which can better reflect the dynamics of the system. Zhang [23] studied the explosion behavior and bifurcation structure caused by pulse shaped explosion (PSE) phenomenon in van der Pol Mathieu system under the action of parameters and external forces. Li [24] explored the explosive oscillation mechanism of van der Pol-Duffing Yeck circuit oscillators with slowly varying parameters and external periodic excitation. Ma [25] studied the different blasting modes and their generation principles in generalized parameter forced van der Pol-Duffing systems.

In recent years, the study of vibration resonance phenomenon has garnered significant attention from researchers across various disciplines. The concept of vibration resonance was initially proposed by Landa and McClintock. In certain nonlinear systems, both low frequency and high frequency signals are simultaneously excited, resulting in a nonlinear relationship between the amplitude of the low frequency signal and the amplitude of the high-frequency signal. This phenomenon, referred to as "resonance" is defined as vibration resonance. Additionally, dual-frequency signals have found widespread applications in physics, communication, control, neuroscience, and other fields [26,27,28,29,30,31,32,33]. Therefore, research on vibration resonance phenomenon holds immense potential value and important significance. Nonlinear systems under multi-frequency excitation exhibit a rich array of phenomena with joint resonance being particularly prominent. For instance, Kwon DukSoo et al. investigated the dual-frequency semi-elliptical shoe antenna employing Mathieu function for Femto-cell network [34]. Jianhua Yang et al., on the other hand, explored vibrational resonance in overdamped systems with fractional potential nonlinearity [35]. Li Jimeng et al.'s work focused on studying a novel adaptive parallel resonance system based on vibrational resonance and stochastic resonance cascade feedback model for its application in rolling bearing fault detection [36].

Based on this, the present study focuses on a class of van der Pol-Duffing subsystems that incorporate fractional fifth power, aiming to investigate their primary sub-harmonic joint resonance. In the first section, an analytical solution is provided for the main sub-harmonic joint resonance involving van der Pol-Duffing oscillators subjected to double-frequency excitation. The approximate analytical solutions exhibit high accuracy across a wide frequency range, thereby confirming the correctness and precision of these solutions. The second section examines the Melnikov condition for chaotic motion under nonlinear stiffness hardening, while the third section analyzes fork bifurcations and vibration resonance within the van der Pol-Duffing subsystem. Finally, numerical simulations are employed in the fourth section to validate the analytical analysis.

2 Main sub-harmonic joint resonance

The purpose of this paper is to study the resonance and bifurcation behavior of fractional-order systems with multi-frequency excitation of the following forms

$$ \ddot{y} + \omega_{0}^{2} y + \left( {\alpha + \beta y^{2} } \right)\dot{y} + \gamma y^{3} + \delta y^{5} + KD_{t}^{p} y\left( t \right) = F_{1} \cos \omega_{1} t + F_{2} \cos \omega_{2} t. $$
(1)

where y is the displacement,\(\omega_{0}\) is the oscillation frequency without damping, \(\alpha ,\beta ,\gamma\) and \(\delta\) describes damping and nonlinear effects [37],\(F_{1}\) and \(F_{2}\) are respectively the amplitudes of two external vibrations, \(\omega_{1}\) and \(\omega_{2}\) are respectively two different frequencies, \(KD_{t}^{p} y(t)\) describes the memory effect. By prefixing the nonlinear term in Eq. (1) with a small parameter \(\varepsilon\), we can obtain

$$ \ddot{y} + \omega_{0}^{2} y + \left( {\varepsilon \alpha + \varepsilon \beta y^{2} } \right)\dot{y} + \varepsilon \gamma y^{3} + \varepsilon \delta y^{5} + \varepsilon KD_{t}^{p} y\left( t \right) = F_{1} \cos \omega_{1} t + F_{2} \cos \omega_{2} t. $$
(2)

Among them, fractional calculus is defined by Caputo type derivative, which is described as

$$ D_{t}^{p} \left[ {y\left( t \right)} \right] = \frac{1}{{\Gamma \left( {1 - p} \right)}}\int_{0}^{t} {\left( {t - \tau } \right)}^{ - p} y^{\prime}\left( \tau \right)d\tau . $$
(3)

where \(\Gamma \left( z \right)\) is the Gamma function.

In order to study the main sub-harmonic joint resonance of the system, it is required that \(\omega_{1} \approx \omega_{0} ,\omega_{2} \approx 3\omega_{0}\) and \(F_{1}\) be a small quantity, that is

$$ F_{1} = \varepsilon f,\omega_{1} = \omega_{0} + \varepsilon \sigma_{1} ,\omega_{2} = 3\omega_{0} + \varepsilon \sigma_{2} ,f = O\left( 1 \right),\sigma_{1} = O\left( 1 \right),\sigma_{2} = O\left( 1 \right). $$

where \(\sigma\) is the introduced tuning factor, thus Eq. (1) can be written

$$ \begin{aligned} \ddot{y} + \omega_{0}^{2} y & = \varepsilon \left[ {f\cos \left( {\omega_{0} t + \varepsilon \sigma_{1} t} \right) - \alpha \dot{y} - \beta y^{2} \dot{y} - \gamma y^{3} - \delta y^{5} - KD_{t}^{p} y\left( t \right)} \right] \\ & + F_{2} \cos \left( {3\omega_{0} t + \varepsilon \sigma_{2} t} \right). \\ \end{aligned} $$
(4)

The multi-scale method is used to study the first approximation solution of the system, two time scales \(T_{0} = t,T_{1} = \varepsilon t\) are introduced, and the solution of Eq. (4) is assumed to have the following form

$$ y\left( {t;\varepsilon } \right) = y_{0} \left( {T_{0} ,T_{1} } \right) + \varepsilon y_{1} \left( {T_{0} ,T_{1} } \right). $$
(5)

By substituting Eq. (5) into Eq. (4) and comparing the same power of \(\varepsilon\), we can get a set of partial differential equations that

$$ D_{0}^{2} y_{0} + \omega_{0}^{2} y_{0} = F_{2} \cos \left( {3\omega_{0} T_{0} + \sigma_{2} T_{1} } \right) $$
(6)
$$ D_{0}^{2} y_{1} + \omega_{0}^{2} y_{1} = - 2D_{0} D_{1} y_{0} + f\cos \left( {\omega_{0} T_{0} + \sigma_{1} T_{1} } \right) - \alpha D_{0} y_{0} - \beta y_{0}^{2} D_{0} y_{0} - \gamma y_{0}^{3} - \delta y_{0}^{5} - KD_{{T_{0} }}^{p} y_{0} . $$
(7)

The solution of Eq. (6) is

$$ y_{0} \left( {T_{0} ,T_{1} } \right) = a\left( {T_{0} } \right)\cos \left[ {\omega_{0} T_{0} + \theta \left( {T_{1} } \right)} \right] + \frac{{F_{2} }}{{\omega_{0}^{2} - \omega_{2}^{2} }}\cos \left( {3\omega_{0} T_{0} + \sigma_{2} T_{1} } \right). $$
(8)

where,\(a\left( {T_{1} } \right)\) and \(\theta \left( {{\text{T}}_{1} } \right)\) are slow-varying amplitude and phase respectively. Write Eq. (8) in the plural form as

$$ y_{0} \left( {T_{0} ,T_{1} } \right) = A\left( {T_{1} } \right)e^{{j\omega_{0} T_{0} }} + Be^{{j\left( {3\omega_{0} T_{0} + \sigma_{2} T_{1} } \right)}} + cc. $$
(9)

where \(A\left( {T_{1} } \right) = \frac{{a\left( {T_{1} } \right)}}{2}e^{{j\theta \left( {T_{1} } \right)}} ,B = \frac{{F_{2} }}{{2\left( {\omega_{0}^{2} - \omega_{2}^{2} } \right)}}\),\(cc\) is the conjugate of all the aforementioned terms. An approximate formula is introduced here [34], and the fractional derivative of \(e^{j\Omega t}\) is approximated by the formula

$$ D_{t}^{p} e^{j\Omega t} \approx \Omega^{p} e^{{j\left( {\Omega t + \frac{p\pi }{2}} \right)}} . $$
(10)

By substituting Eq. (9) into Eq. (7) and calculating \(KD_{{T_{0} }}^{p} y_{0}\) using Eq. (10), the condition of eliminating the perpetual year term can be obtained as

$$ \begin{aligned} & - 2j\omega_{0} D_{1} A - j\alpha \omega_{0} A - 3\gamma A^{2} \overline{A} - 6\gamma AB^{2} - 3\gamma \overline{A}^{2} Be^{{j\sigma_{2} T_{1} }} - 5\delta A^{4} Be^{{ - T_{1} j\sigma_{2} }} \\ & - 10\delta A^{3} \overline{A}^{2} - 60\delta A^{2} \overline{A}B^{2} - 20\delta A\overline{A}^{3} Be^{{T_{1} j\sigma_{2} }} - 30\delta AB^{4} - 30\delta \overline{A}^{2} B^{3} e^{{T_{1} j\sigma_{2} }} \\ & - j\beta A^{2} \overline{A}\omega_{0} - 2j\beta AB^{2} \omega_{0} - j\beta \overline{A}^{2} B\omega_{0} e^{{T_{1} j\sigma_{2} }} - KA\omega_{0}^{p} e^{{j\frac{p\pi }{2}}} + \frac{f}{2}e^{{j\sigma_{1} T_{1} }} = 0. \\ \end{aligned} $$
(11)

By substituting \(A\left( {T_{1} } \right) = \frac{{a\left( {T_{1} } \right)}}{2}e^{{j\theta \left( {T_{1} } \right)}}\) into Eq. (11), and separating the real and imaginary parts, the differential equations satisfied by the slow-varying amplitude \(a\left( {T_{1} } \right)\) and phase \(\theta \left( {T_{1} } \right)\) can be obtained

$$ \begin{aligned} D_{1} a & = - \alpha a - \frac{{\beta a^{3} }}{8} - \beta B^{2} a + \frac{{3\gamma Ba^{2} }}{{4\omega_{0} }}\sin (3\theta - T_{1} \sigma_{2} ) + \frac{{15\delta Ba^{4} }}{{16\omega_{0} }}\sin \left( {3\theta - T_{1} \sigma_{2} } \right) \\ & + \frac{{15\delta B^{3} a^{2} }}{{2\omega_{0} }}\sin \left( {3\theta - T_{1} \sigma_{2} } \right) - \frac{{\beta Ba^{2} }}{4}\cos \left( {3\theta - T_{1} \sigma_{2} } \right) - \frac{a}{2}K\omega_{0}^{p - 1} \sin \frac{p\pi }{2} \\ & - \frac{f}{{2\omega_{0} }}\sin \left( {\theta - T_{1} \sigma_{1} } \right) \\ \end{aligned} $$
(12)
$$ \begin{aligned} aD_{1} \theta & = \frac{{3\gamma a^{3} }}{{8\omega_{0} }} + \frac{{3\gamma B^{2} a}}{{\omega_{0} }} + \frac{{5\delta a^{5} }}{{16\omega_{0} }} + \frac{{15\delta B^{2} a^{3} }}{{2\omega_{0} }} + \frac{{15\delta B^{4} a}}{{\omega_{0} }} + \frac{{3\gamma Ba^{2} }}{{4\omega_{0} }}\cos \left( {3\theta - T_{1} \sigma_{2} } \right) \\ & + \frac{{25\delta Ba^{4} }}{{16\omega_{0} }}\cos \left( {3\theta - T_{1} \sigma_{2} } \right) + \frac{{15\delta B^{3} a^{2} }}{{2\omega_{0} }}\cos \left( {3\theta - T_{1} \sigma_{2} } \right) + \frac{{\beta Ba^{2} }}{4}\sin \left( {3\theta - T_{1} \sigma_{2} } \right) \\ & + \frac{a}{2}K\omega_{0}^{p - 1} \cos \frac{p\pi }{2} - \frac{f}{{2\omega_{0} }}\cos \left( {\theta - T_{1} \sigma_{1} } \right). \\ \end{aligned} $$
(13)

Thus, the first approximation solution of the system can be expressed as

$$ y\left( t \right) = a\cos \left( {\omega_{0} t + \theta } \right) + 2B\cos \omega_{2} t. $$
(14)

where, \(a\) and \(\theta\) are functions of \(T_{1}\), which is determined by Eqs. (12) and (13).

It can be seen from Eqs. (12) and (13) that the necessary condition for the existence of the steady solution of system is that \(\theta - T_{1} \sigma_{1}\) and \(3\theta - T_{1} \sigma_{2}\) are constants, namely \(D_{1} \theta = \sigma_{1} = \sigma_{2} /3\), and then \(\omega_{1} = \omega_{2} /3\). Therefore, the analytical method also needs two excitation frequencies to satisfy this multiple relation to analyze the steady solution of the system.

Introduce tuning factor \(\sigma = \sigma_{1} = \sigma_{2} /3\), and let \(\theta - \sigma T_{1} = \phi\), then Eqs. (12) and (13) become

$$ \begin{aligned} D_{1} a & = - \alpha a - \frac{{\beta a^{3} }}{8} - \beta B^{2} a + \frac{{3\gamma Ba^{2} }}{{4\omega_{0} }}\sin 3\phi + \frac{{15\delta Ba^{4} }}{{16\omega_{0} }}\sin 3\phi \\ & + \frac{{15\delta B^{3} a^{2} }}{{2\omega_{0} }}\sin 3\phi - \frac{{\beta Ba^{2} }}{4}\cos 3\phi - \frac{a}{2}K\omega_{0}^{p - 1} \sin \frac{p\pi }{2} - \frac{f}{{2\omega_{0} }}\sin \phi \\ \end{aligned} $$
(15)
$$ \begin{aligned} aD_{1} \phi & = - \sigma a + \frac{{3\gamma a^{3} }}{{8\omega_{0} }} + \frac{{3\gamma B^{2} a}}{{\omega_{0} }} + \frac{{5\delta a^{5} }}{{16\omega_{0} }} + \frac{{15\delta B^{2} a^{3} }}{{2\omega_{0} }} + \frac{{15\delta B^{4} a}}{{\omega_{0} }} \\ & + \frac{{3\gamma Ba^{2} }}{{4\omega_{0} }}\cos 3\phi + \frac{{25\delta Ba^{4} }}{{16\omega_{0} }}\cos 3\phi + \frac{{15\delta B^{3} a^{2} }}{{2\omega_{0} }}\cos 3\phi + \frac{{\beta Ba^{2} }}{4}\sin 3\phi \\ & + \frac{a}{2}K\omega_{0}^{p - 1} \cos \frac{p\pi }{2} - \frac{f}{{2\omega_{0} }}\cos \phi . \\ \end{aligned} $$
(16)

The corresponding First-order approximation solution becomes

$$ y\left( t \right) = a\cos \left( {\omega_{1} t + \phi } \right) + 2B\cos \omega_{2} t. $$
(17)

where, \(a\) and \(\phi\) are functions of \(T_{1}\), determined by Eqs. (15) and (16).

3 Melnikov condition of chaotic motion

Consider the nonlinear stiffness hardening (i.e.\(\gamma > 0,\delta > 0\)) of the system.

The nonlinear term of Eq. (1) is prefixed with small parameter \(\varepsilon\), and the following coordinate transformation is carried out

$$ F_{1} = \varepsilon \eta_{1} ,F_{2} = \varepsilon \eta_{2} $$

Equation (1) can be written as

$$ \ddot{y} + \omega_{0}^{2} y + \left( {\varepsilon \alpha + \varepsilon \beta y^{2} } \right)\dot{y} + \gamma y^{3} + \delta y^{5} + \varepsilon KD_{t}^{p} y\left( t \right) = \varepsilon \eta_{1} \cos \omega_{1} t + \varepsilon \eta_{2} \cos \omega_{2} t. $$
(18)

Write Eq. (18) as an equation of state

$$ \dot{y}_{1} = y_{2} $$
(19)
$$ \dot{y}_{2} = - \omega_{0}^{2} y_{1} - \gamma y_{1}^{3} - \delta y_{1}^{5} + \varepsilon \left[ {\eta_{1} \cos \omega_{1} t + \eta_{2} \cos \omega_{2} t - \left( {\alpha + \beta y_{1}^{2} } \right)y_{2} - KD_{t}^{p} y_{1} } \right]. $$
(20)

When \(\varepsilon = 0\), the undisturbed system is Hamilton system

$$ \dot{y}_{1} = y_{2} $$
(21)
$$ \dot{y}_{2} = - \omega_{0}^{2} y_{1} - \gamma y_{1}^{3} - \delta y_{1}^{5} . $$
(22)

Its Hamiltonian is

$$ H\left( {y_{1} ,y_{2} } \right) = \frac{1}{2}y_{2}^{2} + \frac{1}{2}\omega_{0}^{2} y_{1}^{2} + \frac{1}{4}\gamma y_{1}^{4} + \frac{1}{6}\delta y_{1}^{6} . $$
(23)

The potential function is

$$ \Delta \left( {y_{1} } \right) = \frac{1}{2}\omega_{0}^{2} y_{1}^{2} + \frac{1}{4}\gamma y_{1}^{4} + \frac{1}{6}\delta y_{1}^{6} . $$
(24)

According to Eqs. (21) and (22), the parametric equation under heterotactic orbit can be written as

$$ y_{1} \left( t \right) = \pm \frac{{h_{1} \sqrt 2 \sinh \left( {\frac{{h_{2} t}}{2}} \right)}}{{\sqrt { - h_{3} + \cosh \left( {h_{2} t} \right)} }} $$
(25)
$$ y_{2} \left( t \right) = \pm \frac{{h_{1} h_{2} \sqrt 2 \left( {1 - h_{3} } \right)\cosh \left( {\frac{{h_{2} t}}{2}} \right)}}{{2\left[ { - h_{3} + \cosh \left( {h_{2} t} \right)} \right]^{3/2} }}. $$
(26)

Thereinto

$$ h_{1} = \sqrt {\frac{{ - \left( {\gamma + \sqrt {\gamma^{2} - \omega_{0} \delta } } \right)}}{2\delta }} $$
$$ h_{2} = h_{1}^{2} \sqrt {2\delta \left( {h_{4}^{2} - 1} \right)} $$
$$ h_{3} = \frac{{5 - 3h_{4}^{2} }}{{3h_{4}^{2} - 1}} $$
$$ h_{4}^{2} = \frac{{\gamma - \sqrt {\gamma^{2} - 4\omega_{0}^{2} \delta } }}{{\gamma + \sqrt {\gamma^{2} - 4\omega_{0}^{2} \delta } }} $$

Write it in vector form

$$ \vec{y}^{0} \left( t \right) = \left[ {\begin{array}{*{20}l} {y_{1} } \\ {y_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} { \pm \frac{{h_{1} \sqrt 2 \sinh \left( {\frac{{h_{2} t}}{2}} \right)}}{{\sqrt { - h_{3} + \cosh \left( {h_{2} t} \right)} }}} \hfill \\ { \pm \frac{{h_{1} h_{2} \sqrt 2 \left( {1 - h_{3} } \right)\cosh \left( {\frac{{h_{2} t}}{2}} \right)}}{{2\left[ { - h_{3} + \cosh \left( {h_{2} t} \right)} \right]^{3/2} }}} \hfill \\ \end{array} } \right]. $$
(27)

Equation (27) is written in vector form as

$$ \begin{aligned} \dot{\vec{y}} & = h\left( {\vec{y}} \right) + \varepsilon g\left( {\vec{y},t} \right) \\ & = \left[ {\begin{array}{*{20}l} {y_{2} } \\ { - \omega_{0}^{2} y_{1} - \gamma y_{1}^{3} - \delta y_{1}^{5} } \\ \end{array} } \right] + \varepsilon \left[ {\begin{array}{*{20}l} 0 \\ {\eta_{1} \cos \omega_{1} t + \eta_{2} \cos \omega_{2} t - \left( {\alpha + \beta y_{1}^{2} } \right)y_{2} - KD_{t}^{p} y_{1} } \\ \end{array} } \right]. \\ \end{aligned} $$
(28)

Then the Melnikov function of this system is

$$ \begin{aligned} M\left( \tau \right) & = \int_{ - \infty }^{ + \infty } {h\left[ {\vec{y}^{0} \left( {t - \tau } \right)} \right]^{{g} \left[ {\vec{y}^{0} \left( {t - \tau } \right),t} \right]}dt} \\ & = \int_{ - \infty }^{ + \infty } {y_{2} \left( {t - \tau } \right)\left\{ {\eta_{1} \cos \omega_{1} t + \eta_{2} \cos \omega_{2} t - \left[ {\alpha + \beta y_{1}^{2} \left( {t - \tau } \right)} \right]y_{2} \left( {t - \tau } \right) - KD_{t}^{p} \left[ {y_{1} \left( {t - \tau } \right)} \right]} \right\}dt} \\ & = \int_{ - \infty }^{ + \infty } { \pm \frac{{h_{1} h_{2} \sqrt 2 \left( {1 - h_{3} } \right)\cosh \left( {\frac{{h_{2} t}}{2}} \right)}}{{2\left[ { - h_{3} + \cosh \left( {h_{2} t} \right)} \right]^{3/2} }}} \\ & \quad \left\{ {\eta_{1} \cos \omega_{1} \left( {t + \tau } \right) + \eta_{2} \cos \omega_{2} \left( {t + \tau } \right) \mp \left[ {\alpha + \beta \frac{{2h_{1}^{2} \sinh^{2} \left( {\frac{{h_{2} t}}{2}} \right)}}{{ - h_{3} + \cosh \left( {h_{2} t} \right)}}} \right]\frac{{h_{1} h_{2} \sqrt 2 \left( {1 - h_{3} } \right)\cosh \left( {\frac{{h_{2} t}}{2}} \right)}}{{2\left[ { - h_{3} + \cosh \left( {h_{2} t} \right)} \right]^{3/2} }} \mp KD_{t}^{p} \left[ {\frac{{h_{1} \sqrt 2 \sinh \left( {\frac{{h_{2} t}}{2}} \right)}}{{\sqrt { - h_{3} + \cosh \left( {h_{2} t} \right)} }}} \right]} \right\}dt \\ \end{aligned} $$
(29)

We can get it by organizing it

$$ \begin{gathered} M(\tau ) = \frac{{2\pi \omega_{1} h_{1} \eta_{1} }}{{h_{2} \sinh \frac{{\pi \omega_{1} }}{{h_{2} }}}}\cos \omega_{1} \tau + \frac{{2\pi \omega_{2} h_{1} \eta_{2} }}{{h_{2} \sinh \frac{{\pi \omega_{2} }}{{h_{2} }}}}\cos \omega_{2} \tau - \frac{{h_{1}^{4} h_{2} \beta }}{{6\left( {1 + h_{3} } \right)^{2} }} - \frac{{h_{1}^{4} h_{2} h_{3}^{2} \beta }}{{12\left( {1 + h_{3} } \right)^{2} }} - \frac{{h_{1}^{4} h_{2} h_{3} \beta }}{{2\left( {1 + h_{3} } \right)^{2} }}\frac{{\arctan \left( {\frac{{1 + h_{3} }}{{\sqrt {1 - h_{3}^{2} } }}} \right)}}{{\sqrt {1 - h_{3}^{2} } }} \\ - \left[ {K\omega_{1}^{p - 1} \sin \left( {\frac{p\pi }{2}} \right) + \alpha } \right]\left( {\frac{{h_{1}^{2} h_{2} h_{3} }}{{2 + 2h_{3} }} + \frac{{h_{1}^{2} h_{2} }}{{2 + 2h_{3} }}\frac{{\arcsin h_{3} + \frac{\pi }{2}}}{{\sqrt {1 - h_{3}^{2} } }} + \frac{{h_{1}^{2} h_{2} h_{3} }}{{1 + 1h_{3} }}\frac{{\arcsin h_{3} + \frac{\pi }{2}}}{{\sqrt {1 - h_{3}^{2} } }} + \frac{{h_{1}^{2} h_{2} }}{{1 + 1h_{3} }}} \right) \\ \end{gathered} $$
(30)

Here \(D_{t}^{p} (y_{1} ) = \omega_{{^{1} }}^{p - 1} \sin \left( {\frac{p\pi }{2}} \right)y_{2} + \omega_{1}^{p} \cos \left( {\frac{p\pi }{2}} \right)y_{1}\). Therefore, according to Melnikov's theorem, the necessary condition for the cross-sectional intersection of alien orbits in system (1) is

$$ \begin{aligned} & \frac{{2\pi \omega _{1} h_{1} \eta _{1} }}{{h_{2} \sinh \frac{{\pi \omega _{1} }}{{h_{2} }}}} + \frac{{2\pi \omega _{2} h_{1} \eta _{2} }}{{h_{2} \sinh \frac{{\pi \omega _{2} }}{{h_{2} }}}}\frac{{h_{1}^{4} h_{2} \beta }}{{6\left( {1 + h_{3} } \right)^{2} }} + \frac{{h_{1}^{4} h_{2} h_{3}^{2} \beta }}{{12\left( {1 + h_{3} } \right)^{2} }} + \frac{{h_{1}^{4} h_{2} h_{3} \beta }}{{2\left( {1 + h_{3} } \right)^{2} }}\frac{{\arctan \left( {\frac{{1 + h_{3} }}{{\sqrt {1 - h_{3}^{2} } }}} \right)}}{{\sqrt {1 - h_{3}^{2} } }} \\ & \quad + \left[ {K\omega _{1}^{{p - 1}} \sin \left( {\frac{{p\pi }}{2}} \right) + \alpha } \right]\left( {\frac{{h_{1}^{2} h_{2} h_{3} }}{{2 + 2h_{3} }} + \frac{{h_{1}^{2} h_{2} }}{{2 + 2h_{3} }}\frac{{\arcsin h_{3} + \frac{\pi }{2}}}{{\sqrt {1 - h_{3}^{2} } }} + \frac{{h_{1}^{2} h_{2} h_{3} }}{{1 + 1h_{3} }}\frac{{\arcsin h_{3} + \frac{\pi }{2}}}{{\sqrt {1 - h_{3}^{2} } }} + \frac{{h_{1}^{2} h_{2} }}{{1 + 1h_{3} }}} \right). \\ \end{aligned} $$
(31)

4 Analysis of system bifurcation characteristics

4.1 Response amplitude gain

This section discusses the following under-damped systems with fractional order

$$ \ddot{y} + \omega_{0}^{2} y + \left( {\alpha + \beta y^{2} } \right)\dot{y} + \gamma y^{3} + \delta y^{5} + KD_{t}^{p} y\left( t \right) = F_{1} \cos \omega_{1} t + F_{2} \cos \omega_{2} t. $$
(32)

where, the system parameter satisfies \(\alpha > 0,\beta > 0,\gamma > 0,\delta > 0,F_{1} < < 1,\omega_{1} < < \omega_{2}\), and the potential function of the system is \(V(y) = \frac{1}{2}\omega_{0}^{2} y^{2} + \frac{1}{4}\gamma y^{4} + \frac{1}{6}\delta y^{6}\). When \(\omega_{0}^{2} < 0\), \(V\left( x \right)\) has the shape of double potential well; When \(\omega_{0}^{2} > 0\), \(V(x)\) has a single well shape.

According to the separation method of fast and slow variables, let \(y = Y + \Psi\), where \(Y\) and \(\psi\) are slow and fast variables with periods of \(2\pi /\omega_{1}\) and \(2\pi /\omega_{2}\), respectively. Substituting \(y = Y + \Psi\) into Eq. (32) yields

$$ \begin{aligned} & \frac{{d^{2} Y}}{{dt^{2} }} + \frac{{d^{2} \Psi }}{{dt^{2} }} + K\frac{{d^{p} Y}}{{dt^{p} }} + K\frac{{d^{p} \Psi }}{{dt^{p} }} + \alpha \frac{dY}{{dt}} + \alpha \frac{d\Psi }{{dt}} + \omega_{0}^{2} Y + \omega_{0}^{2} \Psi \\ & \quad \beta \left( {Y + \Psi } \right)^{2} \frac{dY}{{dt}} + \beta \left( {Y + \Psi } \right)^{2} \frac{d\Psi }{{dt}} + \gamma Y^{3} + 3\gamma Y^{2} \Psi + 3\gamma Y\Psi^{2} + \gamma \Psi^{3} + \delta Y^{5} \\ & \quad + 5\delta Y^{4} \Psi + 10\delta Y^{3} \Psi^{2} + 10\delta Y^{2} \Psi^{3} + 5\delta Y\Psi^{4} + \delta \Psi^{5} = F_{1} \cos \left( {\omega_{1} t} \right) + F_{2} \cos \left( {\omega_{2} t} \right). \\ \end{aligned} $$
(33)

Find an approximate solution to \(\Psi\) in the following linear equation

$$ \frac{{d^{2} \Psi }}{{dt^{2} }} + K\frac{{d^{p} \Psi }}{{dt^{p} }} + \omega_{0}^{2} \Psi = F_{2} \cos \left( {\omega_{2} t} \right). $$
(34)

Let solution \(\Psi\) be of the form

$$ \Psi = \frac{{F_{2} }}{\mu }\cos \left( {\omega_{2} t + \theta } \right). $$
(35)

Using the method of undetermined coefficients we can get

$$ \left\{ {\begin{array}{*{20}l} {\mu^{2} = \left[ {\omega_{0}^{2} + K\omega_{2}^{p} \cos \left( {\frac{p\pi }{2}} \right) - \omega_{2}^{2} } \right]^{2} + \left[ {K\omega_{2}^{p} \sin \left( {\frac{p\pi }{2}} \right)} \right]^{2} } \\ {\theta = \tan^{ - 1} \left[ {\frac{{K\omega_{2}^{p} \sin \left( {\frac{p\pi }{2}} \right)}}{{\omega_{0}^{2} + K\omega_{2}^{p} \cos \left( {\frac{p\pi }{2}} \right) - \omega_{2}^{2} }}} \right]} \\ \end{array} .} \right. $$
(36)

Substitute Eq. (35) into Eq. (33), integrate and average all terms in \(\left[ {0,2\pi /\omega_{2} } \right]\), eliminate the fast variable we can get

$$ \frac{{d^{2} Y}}{{dt^{2} }} + K\frac{{d^{p} Y}}{{dt^{p} }} + \left( {\alpha + \beta } \right)\frac{dY}{{dt}} + \beta Y^{2} \frac{dY}{{dt}} + C_{1} Y + C_{2} Y^{3} + \delta Y^{5} = F_{1} \cos \left( {\omega_{1} t} \right). $$
(37)

where \(C_{1} = \frac{{3\gamma F_{2}^{2} }}{{2\mu^{2} }} + \frac{{5\delta F_{2}^{4} }}{{4\mu^{4} }} + \omega_{0}^{2} ,C_{2} = \frac{{5\delta F_{2}^{2} }}{{\mu^{2} }} + \gamma\).

In Eq. (37), there are still three equilibrium points, namely \(Y_{0}^{*}\) and \(Y_{ \pm }^{*}\), \(Y_{ \pm }^{*} = \pm \left[ {\frac{{ - C_{2} + \sqrt {C_{2}^{2} - 4\delta C_{1} } }}{2\delta }} \right]^{\frac{1}{2}}\) causes the equilibrium point of the fork bifurcation to be \(F_{c} = \left[ {\frac{{ - 3\gamma \mu^{2} + \sqrt {\left( {3\gamma \mu^{2} } \right)^{2} - 20\delta \omega_{0}^{2} \mu^{4} } }}{5\delta }} \right]^{\frac{1}{2}}\), considering the periodic component of the solution of the equation, the constant is eliminated by introducing \(X = Y - Y^{*}\)

$$ \begin{aligned} & \frac{{d^{2} X}}{{dt^{2} }} + K\frac{{d^{p} X}}{{dt^{p} }} + \left[ {\alpha + \beta \left( {X^{2} + 2XY^{*} + Y^{*2} + 1} \right)} \right]\frac{dX}{{dt}} + \left( {C_{1} + 3C_{2} Y^{*2} + 5\delta Y^{*4} } \right)X \\ & \quad + \left( {3C_{2} Y^{*} + 10\delta Y^{*3} } \right)X^{2} + \left( {C_{2} + 10\delta Y^{*2} } \right)X^{3} + 5\delta Y^{*} X^{4} + \delta X^{5} = F_{1} \cos \left( {\omega_{1} t} \right) \\ \end{aligned} $$
(38)

Ignoring the nonlinear term in Eq. (38), a linear equation with respect to \(X\) is obtained

$$ \frac{{d^{2} X}}{{dt^{2} }} + K\frac{{d^{p} X}}{{dt^{p} }} + \omega_{r}^{2} X = F_{1} \cos \left( {\omega_{1} t} \right). $$
(39)

There-into, \(\omega_{r}^{2} = C_{1} + 3C_{2} Y^{*2} + 5\delta Y^{*4}\). Using the coefficient to be determined method, the solution of Eq. (39) is \(X = A_{L} \cos \left( {\omega_{1} t - \phi } \right)\), where

$$ \left\{ {\begin{array}{*{20}l} {A_{L} = \frac{{F_{1} }}{{\sqrt {\left\{ {\omega_{r}^{2} - \left[ {\omega_{1}^{2} - K\omega_{1}^{p} \cos \left( {\frac{p\pi }{2}} \right)} \right]} \right\}^{2} + \left[ {K\omega_{1}^{p} \sin \left( {\frac{p\pi }{2}} \right)} \right]^{2} } }}} \\ {\phi = \tan^{ - 1} \frac{{K\omega_{1}^{p} \sin \left( {\frac{p\pi }{2}} \right)}}{{\omega_{r}^{2} - \left[ {\omega_{1}^{2} - K\omega_{1}^{p} \cos \left( {\frac{p\pi }{2}} \right)} \right]}}} \\ \end{array} } \right.. $$
(40)

Therefore, the response amplitude-frequency gain is

$$ Q = \frac{1}{{\sqrt {\left\{ {\omega_{r}^{2} - \left[ {\omega_{1}^{2} - K\omega_{1}^{p} \cos \left( {\frac{p\pi }{2}} \right)} \right]} \right\}^{2} + \left[ {K\omega_{1}^{p} \sin \left( {\frac{p\pi }{2}} \right)} \right]^{2} } }}. $$
(41)

4.2 Vibration resonance

4.2.1 The bi-stable potential function

When \(\omega_{0}^{2} < 0\), the system (32) has the shape of a bi-stable potential function. When \(F_{2} < F_{c}\), the slow variable moves around the stable equilibrium point \(Y_{ \pm }^{*}\); When \(F_{2} \ge F_{c}\), the slow variable moves around the stable equilibrium point \(Y_{0}^{*}\). In Eq. (41), with \(F_{2}\) as the control parameter, critical point \(F_{VR}\) of resonance should satisfy equation \(\omega_{r}^{2} = \omega_{1}^{2} - K\omega_{1}^{p} \cos \left( {\frac{p\pi }{2}} \right)\) or \(F_{VR} = F_{c}\).\(F_{VR}\) is the real root of equation

$$ \frac{d}{dF}\left\{ {\left\{ {\omega_{r}^{2} - \left[ {\omega_{1}^{2} - K\omega_{1}^{p} \cos \left( {\frac{p\pi }{2}} \right)} \right]} \right\}^{2} + \left[ {K\omega_{1}^{p} \sin \left( {\frac{p\pi }{2}} \right)} \right]^{2} } \right\} = 0 $$
(42)

\(F_{c}\) is the critical bifurcation point where fork bifurcation occurs in equivalent systems. It can be discussed in the following two cases:

Case 1.

When the parameters meet

$$ \omega_{1}^{2} \ge K\omega_{1}^{p} \cos \left( {\frac{p\pi }{2}} \right) - 2\omega_{0}^{2} . $$
(43)

There is only one point \(F_{VR}\) that can cause vibration resonance, namely

$$ F_{VR} = \left[ {\frac{{ - 3\gamma \mu^{2} + \sqrt {9\gamma^{2} \mu^{4} - 20\delta \mu^{4} \left[ {K\omega_{1}^{p} \cos \left( {\frac{p\pi }{2}} \right) - \omega_{1}^{2} + \omega_{0}^{2} } \right]} }}{5\delta }} \right]^{\frac{1}{2}} . $$
(44)

There is a single-peak vibration resonance in this curve \(Q - F_{2}\), and the peak value of the response amplitude gain is

$$ Q_{\max }^{(1)} = \frac{1}{{\left| {K\omega_{1}^{p} \sin \left( {\frac{p\pi }{2}} \right)} \right|}}. $$
(45)

Case 2.

When the parameters meet

$$ 0 < \omega_{1}^{2} \le K\omega_{1}^{p} \cos \left( {\frac{p\pi }{2}} \right). $$
(46)

We get that the critical point for resonance is

$$ F_{VR} = F_{c} . $$
(47)

At this time, single-peak vibration resonance exists in curve \(Q - F_{2}\), and the peak value of amplitude gain of response is

$$ Q_{\max }^{(2)} = \frac{1}{{\sqrt {\left[ {\omega_{1}^{2} - K\omega_{1}^{p} \cos \left( {\frac{p\pi }{2}} \right)} \right]^{2} + \left[ {K\omega_{1}^{p} \sin \left( {\frac{p\pi }{2}} \right)} \right]^{2} } }}. $$
(48)

4.2.2 The case of a single stable potential function

When condition \(\omega_{0}^{2} > 0\) is satisfied, the potential function in Eq. (32) is a single stable potential function. For this case, there is always \(C_{1} > 0,C_{2} > 0\) in Eq. (37) and \(Y^{*} = 0\) in the equivalent potential function. When the parameters meet

$$ \omega_{1}^{2} > K\omega_{1}^{p} \cos \left( {\frac{p\pi }{2}} \right) + \omega_{0}^{2} . $$
(49)

Uni-modal resonance occurs at

$$ F_{VR} = \left[ {\frac{{ - 3\gamma \mu^{2} + \sqrt {9\gamma^{2} \mu^{4} - 20\delta \mu^{4} \left[ {K\omega_{1}^{p} \cos \left( {\frac{p\pi }{2}} \right) - \omega_{1}^{2} + \omega_{0}^{2} } \right]} }}{5\delta }} \right]^{\frac{1}{2}} . $$
(50)

For other cases where Eq. (49) is not satisfied, the curve \(Q - F_{2}\) does not take on a resonant shape, and \(Q\) is a minus function of \(F_{2}\).

5 Numerical simulation

Hydraulic pressing and vertical vibration of plate and strip rolling mill can bring a series of practical effects. Vertical vibration can cause uneven deformation of the plate and strip during the working process of the rolling mill, which may lead to a decrease in surface quality, low dimensional accuracy, and even surface defects and fatigue cracks, affecting product quality and processing accuracy; It will affect the stability and smooth operation of the rolling mill, increase equipment vibration and impact, lead to increased equipment wear, severe component wear, and even equipment failures and shutdowns, thereby affecting production progress and equipment life; It may affect the production efficiency of the rolling mill, leading to production interruption, reduced production speed, and decreased output, thereby increasing production costs and cycles, and affecting the production and operation efficiency of the enterprise. Therefore, the study of hydraulic down vertical vibration in plate and strip rolling mills is of great significance.

In this article, \(\ddot{y}\) represents acceleration, describing the inertia and acceleration response of the system, \(\omega_{0}\) represents the natural vibration frequency of the system, \(\alpha ,\)\(\beta ,\)\(\gamma\) and \(\delta\) respectively represent the damping, stiffness, and nonlinear effects of the system, \(K\) describes the variation of system parameters over time, \(D_{t}^{p} y\left( t \right)\) reflects the inertia and response delay of the system, and \(F_{1} \cos \omega_{1} t\) and \(F_{2} \cos \omega_{2} t\) respectively represent the driving force they are subjected to. We will conduct numerical simulation research on the amplitude frequency response characteristics, chaos, bifurcation, and dynamic behavior of vibration resonance of the system to analyze the stability of rolling mill vibration, improve machining accuracy and efficiency.

5.1 Main sub-harmonic joint resonance

5.1.1 Amplitude-frequency curve comparison

In order to test the correctness and accuracy of the approximate analytical solutions determined by Eqs. (15) and (16), numerical simulation is carried out by using Eq. (2) to calculate the numerical solutions of the response, and the analytical solutions are calculated by using Eqs. (15), (16) and (17) and compared with the numerical solutions.For numerical solutions, the following Eq. (2) calculation format is adopted [34]

$$ x\left( {t_{k} } \right) = y\left( {t_{k - 1} } \right)h^{{q_{1} }} - \sum\limits_{i = 1}^{k} {c_{n}^{{(q_{1} )}} } x\left( {t_{k - 1} } \right) $$
(51)
$$ \begin{gathered} y\left( {t_{k} } \right) = [F_{1} \cos \omega_{1} t + F_{2} \cos \omega_{2} t - \omega_{0}^{2} y - \left( {\varepsilon \alpha + \varepsilon \beta y^{2} } \right)\dot{y} - \\ \varepsilon \gamma y^{3} - \varepsilon \delta y^{5} - \varepsilon KD_{t}^{p} y\left( t \right)]h^{{q_{2} }} - \sum\limits_{i = 1}^{k} {c_{n}^{{(q_{2} )}} } y\left( {t_{k - 1} } \right) \\ \end{gathered} $$
(52)
$$ z\left( {t_{k} } \right) = y\left( {t_{k - 1} } \right)h^{{q_{3} }} - \sum\limits_{i = 1}^{k} {c_{n}^{{(q_{3} )}} } z\left( {t_{k - 1} } \right) $$
(53)

In the above equation,\(q_{1} = q_{2} = 1\), \(q_{3} = 1 - p\), and \(c_{n}^{(q)}\) is the fractional binomial coefficient with the iterative relationship as \(c_{0}^{(q)} = 1\) and \(c_{n}^{(q)} = \left( {1 - \frac{1 + q}{n}} \right)c_{n}^{(q)}\). According to Eq. (10), the initial value of fractional-order term is given by

$$ \left. {D_{t}^{p} \left[ {y\left( t \right)} \right]} \right|_{t = 0} = y\left( 0 \right)\omega_{0}^{p} \cos \left( {\frac{p\pi }{2}} \right). $$
(54)

Take a set of system parameters as follows

$$ \varepsilon = 0.1,\omega_{0} = 2,\alpha = 0.1,\beta = 0.5,\gamma = 3,\delta = 0.5,K = 1,p = 0.6,F_{1} = 0.1,F_{2} = 24 $$

Firstly, the amplitude-frequency characteristics of the steady-state response are calculated. For each given excitation frequency \(\omega_{1}\), simulation duration is set to \(t = 200s\) with a step size of 0.001s. The amplitude of the last 10% response is considered as the steady-state amplitude, denoted as \(\overline{y}\). A comparison between the analytical solution and numerical solution for the amplitude-frequency curve is presented in Fig. 1. Then, a comparison between the displacement time history of numerical and analytical solutions is conducted using an incentive frequency \(\omega_{1} = 2.2,\omega_{2} = 3\omega_{1}\). The initial value of the analytic solution is \(\left( {a_{0} ,\phi_{0} } \right) = \left( {2,0} \right)\) and its corresponding numerical solution \(\left( {y_{0} ,\dot{y}_{0} ,D^{p} y_{0} } \right) = \left( {1.3933,0.0167,1.2413} \right)\), obtained from Eq. (14), are compared by performing simulations under aforementioned conditions. The results are illustrated in Fig. 2. From Fig. 1, it can be observed that approximate analytical solutions exhibit high accuracy across a wide frequency range. Furthermore, Fig. 2 demonstrates that even transient responses can be described more accurately using these solutions. In conclusion, this analysis verifies both correctness and precision of our proposed analytical solution.

Fig. 1
figure 1

Amplitude-frequency characteristic curve

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Fig. 2
figure 2

a,b Displacement time history comparison

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5.1.2 Parameter impact on the system

The influence mechanism of system parameters on the system is analyzed through simulation in this section. Firstly, we investigate the impact of order \(p\) of fractional differentiation on the system. Numerical simulations are conducted for different orders of \(p\), and the amplitude-frequency curves for three orders are obtained, as depicted in Fig. 3.

Fig. 3
figure 3

Influence of fractional differential order \(p\) on the system and other calculation parameters \(\varepsilon = 0.1,\omega_{0} = 2,\alpha = 0.1,\beta = 0.5,\gamma = 3,\delta = 0.5,K = 1,F_{1} = 0.1,F_{2} = 24\)

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When fractional orders of \(p = 0.6\), \(p = 0.5\), and \(p = 0.4\) are selected, the amplitude-frequency curve of the system exhibits a variation curve caused by the fractional order. As depicted in Fig. 4, it can be observed that changes in the order \(p\) primarily impact the movement of jump points while having minimal influence on the overall system behavior. The jump points \(R_{1}\), \(R_{2}\), and \(R_{3}\) in the figure correspond to when \(p\) equals 0.6, 0.5, and 0.4 respectively. With a decrease in fractional differential order \(p\), there is a regular shift of steady-state response jump points towards lower frequencies.

Fig. 4
figure 4

Influence of fractional order term coefficient \(K\) on the system and other calculation parameters are \(\varepsilon = 0.1,\omega_{0} = 2,\alpha = 0.1,\beta = 0.5,\gamma = 3,\delta = 0.5,p = 0.6,F_{1} = 0.1,F_{2} = 24\)

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The influence of the coefficient K of the fractional differential term on the system was subsequently investigated. In this study, a fractional differential order \(p = 0.6\) was maintained, and numerical simulations were conducted with different coefficients for the fractional differential term:\(K = 0.5\),\(K = 1\), and \(K = 1.5\). The resulting change curve in the amplitude-frequency response of the system is presented in Fig. 4. It can be observed from Fig. 4 that varying values of the fractional order differential term coefficient \(K\) also affect the jump point behavior. As \(K\) increases, there is a systematic shift towards lower frequencies in terms of both jump point occurrence and maximum response amplitude until reaching a certain threshold where no jump phenomenon occurs.

Finally, the influence of cubed stiffness \(\beta\) on the system was investigated. In this case, a fractional order differential order \(p = 0.6\) was considered. Regarding cubic stiffness \(\beta\), the discussion is divided into two aspects: stiffness hardening and stiffness softening. As depicted in Fig. 5, it can be observed that different values of cubic stiffness \(\beta\) also affect the jumping point. In the case of stiffness hardening (numerically simulated as \(\beta > 0\), \(\beta = 4\), \(\beta = 3\), and \(\beta = 2\)), the resulting change curve caused by cubic stiffness hardening in the amplitude-frequency curve of the system is shown in Fig. 5a. From Fig. 5a, it can be seen that as \(\beta\) decreases, the jump point of steady-state response consistently shifts towards lower frequencies without altering its maximum response amplitude; however, when \(\beta\) reaches a certain threshold value, no jumping phenomenon occurs anymore. In contrast to this behavior, in cases where there is stiffness softening (numerically simulated as \(\beta < 0\), \(\beta = - 4\), \(\beta = - 3\) and \(\beta = - 2\)), a change curve caused by cubic stiffness softening in the amplitude -frequency curve of the system was obtained and presented in Fig. 5b. It can be observed from Fig. 5b that with an increase in \(\beta\), the jump point of steady-state response regularly moves towards higher frequencies while experiencing a decrease in its maximum response amplitude; furthermore, when \(\beta\) exceeds a certain extent, no jump phenomenon occurs anymore. Furthermore, it can be noted that with changes to cubicstiffness approaching zero, the jumping phenomenon disappears.

Fig. 5
figure 5

a,b Influence of the third stiffness coefficient \(\beta\) on the system and other calculation parameters are \(\varepsilon = 0.1,\omega_{0} = 2,\alpha = 0.1,\beta = 0.5,\delta = 0.5,K = 1,p = 0.6,F_{1} = 0.1,F_{2} = 24\)

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5.2 Chaos Analysis

Select the excitation amplitudes \(F_{1}\) and \(F_{2}\) as control parameters and choose a set of other parameters according to Eq. (30) in order to obtain the Melnikov curve of the system, as depicted in Fig. 6. Fixing \(F_{1}\) at 0.1, designate the excitation amplitude \(F_{2}\) as the bifurcation parameter while selecting another set of parameters accordingly. By means of numerical solution, we can acquire the density distribution map illustrating system bifurcation with respect to \(F_{2}\) within the range [0,25], as shown in Fig. 7. As \(F_{2}\) varies, the system undergoes period doubling bifurcation wherein it transitions from a single period doubling motion to a double period motion. With an increasing number of doublings, the frequency spectrum gradually becomes chaotic leading to unpredictable irregular motion that generates chaos before eventually returning to periodic motion. Through comparing Figs. 6 and 7, it can be concluded that when \(F_{1} = 0.1\) and \(F_{2} = 9\), indicating entry into a chaotic state; moreover, it is observed that crossing Melnikov threshold points are essentially consistent with periodic doubling points thereby validating the accuracy of this method.

Fig. 6
figure 6

a,b Melnikov curves and local plots with excitation amplitudes \(F_{1}\) and \(F_{2}\) as control parameters,other calculation parameters are \(\omega_{0}^{2} = - 1,\alpha = 0.8,\beta = 0.5,\gamma = 1.6,\delta = 2.8,K = 1.4,\omega_{1} = 0.5,\omega_{2} = 6,p = 0.5\)

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Fig. 7
figure 7

Global bifurcation diagram of excitation amplitude \(F_{2}\),other calculation parameters are \(\omega_{0}^{2} = - 1,\alpha = 0.8,\beta = 0.5,\gamma = 1.6,\delta = 2.8,K = 1.4,F_{1} = 0.1,\omega_{1} = 0.5,\omega_{2} = 6,p = 0.5\)

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The phase trajectories and Poincare cross-sections of the system at \(F_{2}\) values of 3, 6.5, 9, 14, 15, and 20 are depicted in Fig. 8a–f. It can be observed that for \(F_{2} = 3\) and \(F_{2} = 6.5\), the phase trajectory exhibits a stable limit cycle while the Poincare cross-section has one and two attraction points respectively. Hence, the system demonstrates classical single period and double period motion without chaotic behavior as shown in Fig. 8a–b. As \(F_{2}\) increases further to reach \(F_{2} = 9\), the system enters a chaotic state as illustrated in Fig. 8d. A brief multi-period stable state emerges when \(F_{2}\) equals to approximately 14.23 according to Fig. 7 as shown in Fig. 8f. However, with continued increase of \(F_{2}\) beyond this point, it eventually transitions into a fully chaotic state. When \(F_{2}\) reaches 20, the phase trajectory no longer remains confined within a bounded region but instead becomes entangled and rotates within a Mobius ring pattern repeatedly. The Poincare cross-section is no longer represented by a finite set of points or closed curve; rather it exhibits clear fractal structure characterized by complexity and irregularity along with periodic motion patterns present within it. Therefore, based on these observations presented in Fig. 8f, it can be concluded that the system has already entered into a chaotic state.

Fig. 8
figure 8

a–f Phase trajectory and Poincare cross-section,other calculation parameters are \(\omega_{0}^{2} = - 1,\alpha = 0.8,\beta = 0.5,\gamma = 1.6,\delta = 2.8,K = 1.4,F_{1} = 0.1,\omega_{1} = 0.5,\omega_{2} = 6,p = 0.5\)

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The bifurcation diagrams of the safe basin for \(F_{2}\) values of 3, 6.5, 9, 15, 20, and 23 are depicted in Fig. 9a–f, respectively. Through analysis of the bifurcation and global bifurcation of the safety basin, it can be concluded that fractal erosion of the safety basin initiates at \(F_{2} = 9\). Moreover, both the threshold \(F_{2} \ge 9\) for chaos to occur in the global bifurcation and its analytical conditions obtained by Melnikov exhibit a similar consistency.

Fig. 9
figure 9

a–f Safe basin bifurcation with \(F_{2}\) changes, other calculation parameters are \(\omega_{0}^{2} = - 1,\alpha = 0.8,\beta = 0.5,\gamma = 1.6,\delta = 2.8,K = 1.4,F_{1} = 0.1,\omega_{1} = 0.5,\omega_{2} = 6,p = 0.5\)

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5.3 Analysis of system bifurcation characteristics

5.3.1 Pitchfork bifurcation

The fork bifurcation caused by the control parameter \(F_{2}\) with different fractional differential orders \(p = 0.3\), \(p = 0.6\), \(p = 0.9\), \(p = 1.2\), and \(p = 1.5\) was investigated in this study. Numerical simulations were conducted for \(p = 1.5\) and \(p = 1.8\) to analyze the bifurcation induced by the control parameter \(F_{2}\) (Fig. 10a–f). It can be observed from Fig. 10 that as \(F_{2}\) increases, the equilibrium point \(X_{1.2}^{*}\) disappears while the unstable equilibrium point \(X_{0}^{*}\) transforms into a stable one, indicating a sub-critical bifurcation due to the control parameter \(F_{2}\) influence. Additionally, an increase in fractional differential order \(p\) leads to a rightward shift of the vanishing point \(X_{1.2}^{*}\).

Fig. 10
figure 10

Sub-critical fork bifurcation caused by control parameter \(F_{2}\) with different fractional differential order p, other calculation parameters are \(\omega_{0}^{2} = - 1,\gamma = 2.5,\delta = 2.5,K = 1,\omega_{2} = 6\)

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In order to investigate the fork bifurcation induced by varying the fractional differential order \(p\) under different values of the control parameter \(F_{2}\), numerical simulations were conducted for \(F_{2} = 35\),\(F_{2} = 40\),\(F_{2} = 45\),\(F_{2} = 20\),\(F_{2} = 25\), and \(F_{2} = 30\) respectively. The results are shown in Fig. 11g–l, revealing the occurrence of fork bifurcation due to fractional differentiation. It can be observed from Fig. 11 that as the value of \(F_{2}\) increases, the stable equilibrium point \(X_{0}^{*}\) transitions into an unstable equilibrium point while a new stable equilibrium point \(X_{1.2}^{*}\) emerges; thus indicating that the control parameter \(F_{2}\) induces a supercritical fork bifurcation. Furthermore, with increasing values of the control paramete \(F_{2}\), the stable equilibrium point \(X_{1.2}^{*}\) also shifts towards rightwards. Both Figs. 10 and 11 demonstrate distinct fork bifurcation behaviors resulting from variations in both parameters: \(F_{2}\) and fractional differential order \(p\).

Fig. 11
figure 11

Supercritical fork bifurcation caused by fractional differential order p with different control parameters \(F_{2}\) other calculation parameters \(\omega_{0}^{2} = - 1,\gamma = 2.5,\delta = 2.5,K = 1,\omega_{2} = 6\)

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5.3.2 Vibration resonance

The response amplitude gain is numerically simulated and the analytical results are validated. Figure 12a illustrates the relationship between the response amplitude gain \(Q\), the control variable \(F_{2}\), and the fractional differential order \(p\). As the fractional differential order \(p\) varies, the peak shape of the \(Q - F_{2}\) curve transitions from a single peak to a double peak and then back to a single peak. In Fig. 12b, for a fractional differential order of \(p = 0.45\) that satisfies Eq. (46), the response amplitude gain \(Q\) reaches its maximum value \({\text{Q}}_{{{\text{max}}}}^{(2)}\) at the bifurcation point \(F_{2} = F_{c}\). Figure 12c demonstrates that when taking a fractional differential order of \(p = 1\) without meeting conditions stated in Eqs. (43) and (46), bimodal vibration resonance occurs in the \(Q - F_{2}\) curve. By considering a fractional differential order of \(p = 1.6\) as shown in Fig. 12d while satisfying Eq. (43), it can be observed that the response amplitude gain \(Q\) reaches a local minimum at \(F_{c}\) and peaks at \(F_{VR}\). Within \([0,F_{c} ]\),\(Q\) decreases with increasing \(F_{2}\); within \([F_{c} ,F_{VR} ]\),\(Q\) increases with increasing \(F_{2}\); beyond \(F_{VR}\), \(Q\) decreases with increasing \(F_{2}\).

Fig. 12
figure 12

In the case of bi-stable potential function, a the analytical solution of the response amplitude gain \(Q\) and the relationship between \(F_{2}\) and \(p\), b ~d: the relationship between the response amplitude gain \(Q\) and \(F_{2}\) when \(p\) takes different values, and other calculation parameters are \(\omega_{0}^{2} = - 1,\omega_{1} = 1,\omega_{2} = 5,\alpha = 0.1,\beta = 0.1,\gamma = 8,\delta = 6,K = 1.4,F_{1} = 0.05\)

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The analytical and numerical solutions for the amplitude gain of the response are presented in Fig. 13. In Fig. 13a, as the fractional differential order \(p\) increases, the relationship between response amplitude gain \(Q\) and \(F_{2}\) transitions from monotone decreasing to non-monotone behavior. In Fig. 13b and c, we consider fractional differential orders \(p = 0.5\) and \(p = 1\) respectively. However, these parameter values do not satisfy the conditions stated in Eq. (49), resulting in no occurrence of vibration resonance phenomenon. Finally, in Fig. 13d, when the fractional differential order is set to \(p = 1.5\), satisfying Eq. (49), a vibration resonance occurs at \(F_{2} = F_{VR}\).

Fig. 13
figure 13

In the case of mono-stable potential function, a the relationship between the analytical solution of response amplitude gain \(Q\) and \(F_{2}\) and \(p\), b ~ d: the relationship between response amplitude gain \(Q\) and \(F_{2}\) when \(p\) takes different values, and other calculation parameters are \(\omega_{0}^{2} = 1,\omega_{1} = 1,\omega_{2} = 5,\alpha = 0.1,\beta = 0.1,\gamma = 8,\delta = 6,K = 2,F_{1} = 0.05\)

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In the case of a bi-stable potential function, the impact of the fractional differential order \(p\) on the gain in response amplitude \(Q\) is calculated, as illustrated in Fig. 14a. The functional relationship between \(Q\) and both the control parameter \(F_{2}\) and fractional differential order \(p\) is provided. The fractional differential order \(p\) is considered as a controllable variable. When different values are assigned to \(F_{2}\)(i.e., \(F_{2} = 5\),\(F_{2} = 10\),\(F_{2} = 15\),\(F_{2} = 20\),\(F_{2} = 25\), and \(F_{2} = 30\)), the relationships shown in Fig. 14b–c are obtained. As \(F_{2}\) increases, the monotonic relationship between \(Q\) and \(p\) transitions from being strictly monotonic to non-monotonic and then back to being strictly monotonic again. This demonstrates that the nature of this relationship depends on the magnitude of high-frequency signal amplitude represented by \(F_{2}\).

Fig. 14
figure 14

In the case of bi-stable potential function, a the relationship between the analytical solution of response amplitude gain \(Q\) and \(F_{2}\) and \(p\), b ~c: the relationship between response amplitude gain \(Q\) and fractional differential order \(p\) when \(F\) values are different calculate parameters \(\omega_{0}^{2} = - 1,\omega_{1} = 1,\omega_{2} = 5,\alpha = 0.1,\beta = 0.1,\gamma = 8,\delta = 6,K = 1.4,F_{1} = 0.1\)

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The vibration resonance induced by different fractional differential orders \(p\) is investigated in this study, as depicted in Fig. 15. The peak value of the response amplitude gain is expressed by Eqs. (45) and (48). It can be observed from Fig. 15 that as the fractional differential order \(p\) increases, the vibration resonance mode undergoes a transition from single-peak to double-peak and then back to single-peak resonance. Simultaneously, the fractional differential order \(p\) also influences the resonance position \(F_{VR}\) and the maximum peak \(Q_{\max }\). Parameters such as resonance position, response amplitude gain, and vibration resonance mode of the aforementioned research system contribute to optimizing the design of physical models. For instance, in mechanical structure vibration control, reducing the amplitude gain of resonant response can effectively mitigate structural vibrations, thereby preventing equipment failure or enhancing system performance.

Fig. 15
figure 15

Vibration resonance caused by difference of fractional differential order \(p\) and other calculated parameters are \(\omega_{0}^{2} = - 1,\omega_{1} = 1,\omega_{2} = 5,\alpha = 0.1,\beta = 0.1,\gamma = 8,\delta = 6,K = 1.4,F_{1} = 0.05\)

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6 Conclusion

The nonlinear dynamic equation of a class of van der Pol-Duffing system with quintic oscillator under dual frequency excitation is established in this paper. The analytical solution is analyzed using the multi-scale method, Melnikov method, and fast and slow variable separation method to calculate the chaotic conditions and analyze the vibration and resonance of the system. Successively, this paper analyzes the effects of fractional order, fractional differential coefficient, and cubic stiffness on co-amplitude-frequency curves of the first problem. By comparing the amplitude -frequency curves of the analytical solution and numerical solution, it is observed that there exists a jumping phenomenon in the main sub-harmonic joint resonance of the van der Pol-Duffing oscillator. The decrease in fractional order, increase in fractional differential coefficient, and decrease in cubic stiffness result in regular movement of the jumping point towards lower frequencies. Furthermore, an increase in fractional differential coefficient or zero value for cubic stiffness eliminates this jumping phenomenon. In regards to the second problem which examines chaos within the system, as depicted by Fig. 7, bifurcation caused by excitation amplitude \(F_{2}\) leads to period-doubling bifurcation. With each doubling event occurring more frequently over time, eventually leading to chaotic behavior before returning back to periodic motion. Lastly, we explore bifurcation and vibration resonance under different conditions for our third problem. The results indicate that variations in the excitation coefficient lead to sub-critical bifurcation, while changes in the fractional order result in supercritical bifurcation. Moreover, alterations in the excitation coefficient transform the relationship between response amplitude gain and fractional order from a non-monotonic function to a monotonic function and then back to a non-monotonic function. Additionally, differences in fractional order affect the resonance location, response amplitude gain, and vibration resonance mode of the system. As the fractional order decreases, the resonance location gradually shifts towards higher values on the frequency axis, and the vibration resonance mode transitions from single-peak vibration resonance to bimodal vibration resonance before returning to single-peak vibration resonance. Furthermore, with varying fractional orders, there is an initial decrease followed by gradual increase in peak value of response amplitude gain. Through this analysis, it can be concluded that different parameters can be controlled to manage and eliminate phenomena such as system jumping behavior, bifurcation occurrences, and vibration resonances. This approach enhances model accuracy and stability while providing theoretical support for designing more reliable and efficient dynamic systems.