Abstract
In this paper, we study the number of limit cycles of a class of quartic Liénard systems with polynomial perturbations of degree n, \(20\le n\le 24\). Let H(n, 4) denote the maximal number of limit cycles of Liénard system \({\dot{x}}=y,\, {\dot{y}}=-g(x)-\varepsilon f(x)y\), where \(\varepsilon \ge 0\) is a small parameter, f(x), g(x) are polynomials in x and \(\deg f=n\), \(\deg g=m\). We obtain five better lower bounds of H(n, 4) for \(20\le n\le 24\), which greatly improve the existing results.
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Acknowledgments
The project was supported by National Natural Science Foundation of China (11571090, 11101118), Natural Science Foundation of Hebei Province (A2012205074) and Science Foundation of Hebei Normal University (L2011B01). The authors would like to thank the editor and reviewers for their valuable comments and suggestions, which greatly improve the presentation of this paper.
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Appendix
Appendix
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1.
\(\delta _{20}^*=(a_0^*,\dots ,a_3^*,0,a_5^*,\dots ,a_8^*,0,a_{10}^*,\dots ,a_{13}^*,0,\) \(a_{15}^*,\dots ,a_{18}^*,0,a_{20})\), where
$$\begin{aligned}&a_0^*=0,\\&a_1^*=- 21318.0604324328412532274239106\,a_{{20}},\\&a_2^*= 114011.145914969967385579906941\,a_{{20}},\\&a_3^*=- 173549.546643392826627273550805\,a_{{20}},\\&a_5^*= 326771.242306230057428053157970\,a_{{20}},\\&a_6^*=- 608489.921888034105185035441300\,a_{{20}},\\&a_7^*= 616755.444203830151182066142852\,a_{{20}},\\&a_8^*=- 301315.973806968855186022911247\,a_{{20}},\\&a_{10}^*=98303.7871028734698384939190631\,a_{{20}},\\&a_{11}^*=- 85265.2049634603586986959830478\,a_{{20}},\\&a_{12}^*= 46373.4799399055285103784992794\,a_{{20}},\\&a_{13}^*=- 13300.2603261692754459843728864\,a_{{20}},\\&a_{15}^*=1648.57325620460711482565821829\,a_{{20}},\\&a_{16}^*=- 867.352241550502709346464586178\,a_{{20}},\\&a_{17}^*=291.646481252879500372907979838\,a_{{20}},\\&a_{18}^*=- 49.9989032578958541840445211182\,a_{{20}}. \end{aligned}$$ -
2.
\(\delta _{21}^*=(a_0^*,\dots ,a_3^*,0,a_5^*,\dots ,a_8^*,0,a_{10}^*,\dots ,a_{13}^*,0,\) \(a_{15}^*,\dots ,a_{18}^*,0, a_{20}^*,a_{21})\), where
$$\begin{aligned}&a_0^*=0,\\&a_1^*=162150.853297587914262470460314\,a_{{21}},\\&a_2^*= - 867199.183794196425663602226662\,a_{{21}},\\&a_3^*=1320064.13282112808169539942405\,a_{{21}},\\&a_5^*=- 2485509.30045630063283495496896\,a_{{21}},\\&a_6^*=4628340.19202892555340450692826\,a_{{21}},\\&a_7^*= - 4691227.90018677121080080796830\,a_{{21}},\\&a_8^*= 2291938.49131512648517110662970\,a_{{21}},\\&a_{10}^*=- 747987.385428451731464214392635\,a_{{21}},\\&a_{11}^*=649241.426175244298933278879690\,a_{{21}},\\&a_{12}^*= - 353650.664563661267614856552788\,a_{{21}},\\&a_{13}^*=101796.365731270534398572832549\,a_{{21}},\\&a_{15}^*=- 12998.4349348140403957017396799\,a_{{21}},\\&a_{16}^*= 7084.00410184684354502096733874\,a_{{21}},\\&a_{17}^*=- 2481.48063426816712649288457994\,a_{{21}},\\&a_{18}^*=450.690680282000573788877176248\,a_{{21}},\\&a_{20}^*=- 12.8061529482360835142654870435\,a_{{21}}. \end{aligned}$$ -
3.
\(\delta _{22}^*=(a_0^*,\dots ,a_3^*,0,a_5^*,\dots ,a_8^*,0,a_{10}^*,\dots ,\) \(a_{13}^*,0,a_{15}^*,\dots ,a_{18}^*,0, a_{20}^*,a_{21}^*,a_{22})\), where
$$\begin{aligned}&a_0^*=0,\\&a_1^*=- 831866.271572912296896715258323\,a_{{22}},\\&a_2^*=4448905.06003121615176894304027\,a_{{22}},\\&a_3^*=- 6772192.72236487595894435846335\,a_{{22}},\\&a_5^*=12751159.2076311909764969835906\,a_{{22}},\\&a_6^*=- 23744308.7739395490436273790461\,a_{{22}},\\&a_7^*= 24066931.1513193240610400762475\,a_{{22}},\\&a_8^*= - 11758092.7077670827587730931060\,a_{{22}},\\&a_{10}^*= 3837305.12070261853266974367342\,a_{{22}},\\&a_{11}^*=- 3330754.12526237265404871436386\,a_{{22}},\\&a_{12}^*= 1814444.44549217981744219235528\,a_{{22}},\\&a_{13}^*=- 522451.805988802594961667325535\,a_{{22}},\\&a_{15}^*=67097.4221011585639705155696440\,a_{{22}},\\&a_{16}^*= - 36970.6591133074643673558769376\,a_{{22}},\\&a_{17}^*= 13197.9382609075255838813683705\,a_{{22}},\\&a_{18}^*=- 2480.84116329836201760544413679\,a_{{22}},\\&a_{20}^*= 90.4224190528554906586144065030\,a_{{22}},\\&a_{21}^*=- 13.8607854473508261055753021643\,a_{{22}}. \end{aligned}$$ -
4.
\(\delta _{23}^*=(a_0^*,\dots ,a_3^*,0,a_5^*,\dots ,a_8^*,0,a_{10}^*,\dots ,\) \(a_{13}^*,0,a_{15}^*,\dots ,a_{18}^*,0, a_{20}^*,a_{21}^*,a_{22}^*,a_{23})\), where
$$\begin{aligned}&a_0^*=0,\\&a_1^*=1865551.24483979783699885343007\,a_{{23}},\\&a_2^*= - 9977156.97398877850734186811728\,a_{{23}},\\&a_3^*=15187384.0761313893129823160543\,a_{{23}},\\&a_5^*=- 28595871.1220563970465268861918\,a_{{23}},\\&a_6^*=53249209.2730054201341195484786\,a_{{23}},\\&a_7^*= - 53972708.8817114733396310295925\,a_{{23}},\\&a_8^*= 26368770.6120695495321288948130\,a_{{23}},\\&a_{10}^*=- 8605435.78480191818093671730375\,a_{{23}},\\&a_{11}^*= 7469357.21410967127392641705277\,a_{{23}},\\&a_{12}^*= - 4069068.71856143046166772866878\,a_{{23}},\\&a_{13}^*=1171905.48533885523779570202556\,a_{{23}},\\&a_{15}^*=- 151283.904982231202211905962712\,a_{{23}},\\&a_{16}^*= 84291.3850650009097699069459751\,a_{{23}},\\&a_{17}^*=- 30734.9757997159890829386767773\,a_{{23}},\\&a_{18}^*=6023.05278893108867666421038019\,a_{{23}},\\&a_{20}^*=- 298.959012037341821283616853025\,a_{{23}},\\&a_{21}^*=78.9599811490577454656732700005\,a_{{23}},\\&a_{22}^*=- 12.9824157823149234105535552428\,a_{{23}}. \end{aligned}$$
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Yang, J., Zhou, L. Limit cycle bifurcations in a kind of perturbed Liénard system. Nonlinear Dyn 85, 1695–1704 (2016). https://doi.org/10.1007/s11071-016-2787-0
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DOI: https://doi.org/10.1007/s11071-016-2787-0