Limit cycle bifurcations in a kind of perturbed Liénard system

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.

Appendix

• 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

  $$\begin{aligned}&a_0^*=0,\\&a_1^*=1865551.24483979783699885343007\,a_{{23}},\\&a_2^*= - 9977156.97398877850734186811728\,a_{{23}},\\&a_3^*=15187384.0761313893129823160543\,a_{{23}},\\&a_5^*=- 28595871.1220563970465268861918\,a_{{23}},\\&a_6^*=53249209.2730054201341195484786\,a_{{23}},\\&a_7^*= - 53972708.8817114733396310295925\,a_{{23}},\\&a_8^*= 26368770.6120695495321288948130\,a_{{23}},\\&a_{10}^*=- 8605435.78480191818093671730375\,a_{{23}},\\&a_{11}^*= 7469357.21410967127392641705277\,a_{{23}},\\&a_{12}^*= - 4069068.71856143046166772866878\,a_{{23}},\\&a_{13}^*=1171905.48533885523779570202556\,a_{{23}},\\&a_{15}^*=- 151283.904982231202211905962712\,a_{{23}},\\&a_{16}^*= 84291.3850650009097699069459751\,a_{{23}},\\&a_{17}^*=- 30734.9757997159890829386767773\,a_{{23}},\\&a_{18}^*=6023.05278893108867666421038019\,a_{{23}},\\&a_{20}^*=- 298.959012037341821283616853025\,a_{{23}},\\&a_{21}^*=78.9599811490577454656732700005\,a_{{23}},\\&a_{22}^*=- 12.9824157823149234105535552428\,a_{{23}}. \end{aligned}$$