1 Introduction
2 Description of the problem
3 The perturbation methodology
4 Resonance categorizations and modulating equations
5 Steady-state solutions
6 Examination of the stability
Figure | PFP | CFP | \(\sigma_{2}\) |
---|---|---|---|
Figure 18a | (− 0..391011, 0.105832) (− 0.184238, 1.25089) | (− 0.184238, 1.25089) | 0.192984 |
Figure 18b | (− 0.403174, 0.110184) (− 0.200626, 1.0473) (− 0.0918118, 0.117015) (0.00340044, 0.452913) | (− 0.200626, 1.0473) | 0.192984 |
Figure 18c | (− 0.500479, 0.0998696) (− 0.384929, 0.179824) (− 0.293706, 0.207501) (− 0.200626, 0.849878) (− 0.10185, 0.113436) (0.0999664, 0.315835) | (− 0.200626, 0.849878) | 0.192984 |
Figure | PFP | CFP | \(\sigma_{2}\) |
---|---|---|---|
Figure 19a | (− 0.308099, 0.07366890) (− 0.186213, 3.85488) (− 0.0914139, − 1.46375) (0.0033857, 2.23436) | (− 0.186213, 3.85488) | 0.192984 |
Figure 19b | (− 0.289412, 0.00585496) (− 0.185354, 3.01606) (− 0.10731, − 0.0283519) | (− 0.185354, 3.01606) | 0.192984 |
Figure 19c | (− 0.397092, 0.074943) (− 0.299787, 0.0272784) (− 0.198361, 1.61391) (− 0.0812955, 0.0547762) | (− 0.198361, 1.61391) | 0.192984 |
Figure | PFP | CFP | \(\sigma_{2}\) |
---|---|---|---|
Figure 20a | (− 0.409255, 0.109994) (− 0.196401, 0.790384) (− 0.122032, 0.829729) (0.00340044, 0.101236) (0.112214, 0.258865) | (− 0.196401, 0.790384) | 0 |
Figure 20b | (− 0.299787, 0.123756) (− 0.202482, 1.16459) (− 0.0102013, 0.102463) (0.15302, 0.193026) | (− 0.202482, 1.16459) | 0 |
Figure 20c | (− 0..603865, 0.099717) (− 0..500479, 0.121885) (− 0..397092, 0.103128) (− 0..293706, 0.128706) (− 0..184238, 0.474875) (− 0.108577, 0.489359) (0.0057854, 0.103902) (0.207602, 0.178228) | (− 0.184238, 0.474875) | 0 |
Figure | PFP | CFP | \(\sigma_{2}\) |
---|---|---|---|
Figure 21a | (− 0.224376, 0.282419) (− 0.10731, 1.39554) (0.0227627, 0.230646) | (− 0.224376, 0.282419) | 0 |
Figure 21b | (− 0.198361, -0.00233394) (− 0.10731, 1.05902) (0.00975549, 0.204759) | (− 0.198361, -0.00233394) | 0 |
Figure 21c | (− 0.494397, 0.0714494) (.-0.196401, 0.527472) | (.-0.196401, 0.527472) | 0 |
Figure | PFP | CFP | \(\sigma_{2}\) |
---|---|---|---|
Figure 22a | (− 0.397092, 0.110284) (− 0.281543, 0.237859) (− 0.190319, 1.3615) (0.0999664, 0.199994) | (− 0.190319, 1.3615) | − 0.0514914 |
Figure 22b | (− 0.282236, 0.117462) (0.214227, 0.85852) (0.112214, 0.164602) | (0.214227, 0.85852) | − 0.0514914 |
Figure 22c | (− 0.403174, 0.100322) (− 0.293706, 0.124925) (− 0.10185, 0.461135) (0.0170022, 0.117628) (0.23463, 0.143168) | (− 0.196401, 0.358003) | − 0.0514914 |
Figure | PFP | CFP | \(\sigma_{2}\) |
---|---|---|---|
Figure 23a | (− 0.299787, 0.0782844) (− 0.196401, − 0.321224) (− 0.09430281, 1.9292) (− 0.003251, -0.528545) | (− 0.196401, − 0.321224) | − 0.0514914 |
Figure 23b | (− 0.185354, − 0.00930659) (− 0.0812955, 0.71634) (0.00975549, 0.0874464) | (− 0..185354, − 0.00930659) | − 0.0514914 |
Figure 23c | – | (− 0.0422737, − 0.0334947) | − 0.0514914 |
Figure | PFP | CFP | \(\sigma_{2}\) |
---|---|---|---|
Figure 24a | (− 0.404652, 0.103169) (− 0.187024, 1.15527) (− 0.39105, 0.10923) (− 0.200626, 1.25516) (− 0.472661, 0.106736) (− 0.187024, 1.36248) | (− 0.187024, 1.15527) (− 0.200626, 1.25516) (− 0.187024, 1.36248) | 0.192984 |
Figure 24b | (− 0.294556, 0.037748) (− 0.186213, 3.4868) (− 0.0914139, − 1.98857) (0.0304713, 2.10718) (− 0.321641, − 0.00943457) (− 0.199756, 3.8133) (− 0.104957, − 1.5053) (0.0169285, 2.1097) (− 0.276405, − 0.0130093) (− 0.198361, 4.2916) (− 0.0812955, 1.88509) | (− 0.186213, 3.4868) (− 0.199756, 3.81333) (− 0.198361, 4.2916) | 0.192984 |
Figure 24c | (− 0.39105, 0.10923) (− 0.200626, 125516) (− 0.350245, 0.109231) (− 0.200626, 1.25054) (− 0.363847, 0.0999894) (− 0.187024, 1.25054) | (− 0.200626, 1.25516) (− 0.200626, 1.25554) | 0.192984 |
Figure 24d | (− 0.321641, − 0.00943457) (− 0.199756, 3.8133) (− 0.104957, − 1.5053) (0.0169285, 2.1097) (− 0.281013, − 0.0509863) (− 0.186213, 3.81333) (− 0.0914139, − 1.54685) (0.0101351, 2.2504) (− 0.281013, 0.0321172) (− 0.186213, 3.85488) (− 0.0914139, − 1.46375) (0.0033857, 2.23436) | (− 0.199756, 3.81333) (− 0.186213, 3.81333) (− 0. 186213, 3.85488) | 0.192984 |
Figure 24e | (− 0.39105, 0.10923) (− 0.200626, 125516) (− 0.380849, 0.105765) (− 0.197225, 1.25632) (− 0.38552, 0.105379) (− 0.192776, 1.2497) | (− 0.200626, 1.25516) (− 0.197225, 1.25632) (− 0.192776, 1.2497) | 0.192984 |
Figure 24f | (− 0.321641, − 0.00943457) (− 0.199756, 3.8133) (− 0.104957, − 1.5053) (0.0169285, 2.1097) (− 0.278566, − 0.144181) (− 0.197528, 3.83945) (− 0.106361, − 1.53732) (0.00506483, 2.22331) (− 0.268436, 0.016615) (− 0.197528, 3.87053) (0.0962318, − 1.50624) (0.0151945, 2.22331) | (− 0.199756, 3.81333) (− 0.197528, 3.83945) (− 0.197528, 3.87053) | 0.192984 |
Figure | PFP | CFP | \(\sigma_{2}\) |
---|---|---|---|
Figure 25a | (− 0.268634, 0.120127) (− 0.200626, 1.25403) (0.00340044, 0.101539) (0.112214, 0.21307) (− 0.404652, 0.110509) (− 0.0918118, 0.833747) (0.00340044, 0.101236) (0.112214, 0.258865) (− 0.486262, 0.103274) (− 0.105413, 0.94891) (0.00340044, 0.0997502) (0.0986126, 0.367535) | (− 0.200626, 1.25403) (− 0.200626, 0.790477) (− 0.187024, 0.772735) | 0 |
Figure 25b | (− 0.302419, 0.0672743) (− 0.198361, 0.300746) (− 0.0812955, 0.868745) (0.00975546, 0.200853) (− 0.0943028, 1.42143) (0.022767, 0.256532) (− 0.289412, − 0.00240918) (− 0.0812955, 1.86145) (0.0097546, 0.204686) | (− 0.198361, − 0.300746) (− 0.185354, 0.385965) (− 0.172347, 1.3696) | 0 |
Figure 25c | (− 0.404652, 0.110509) (− 0.0918118, 0.833747) (0.00340044, 0.101236) (0.112214, 0.258865) (− 0.377448, 0.1074108) (− 0.105413, 0.836838) (0.00340044, 0.0981456) (0.0986126, 0.0252684) (− 0.363847, 0.113599) (− 0.0918117, 0.833747) (0.0170022, 0.104377) (0.112214, 0.255775) | (− 0.200626, 0.790477) (− 0.200626, 0.78495) (− 0.187024, 0.793567) | 0 |
Figure 25d | (− 0.0943028, 1.42143) (0.022767, 0.256532) (− 0.0943028, 1.39554) (0.0227627, 0.256532) (− 0.10731, 0.142143) (0.00650365, 0.237117) | (− 0.185354, 0.385965) (− 0.1853541, 0.360079) (− 0.185354, 0.308306) | 0 |
Figure 25e | (− 0.404652, 0.110509) (− 0.0918118, 0.833747) (0.00340044, 0.101236) (0.112214, 0.258865) (− 0.385552, 0.108517) (− 0.101461, 0.837064) (0.0101461, 0.1016) (0.11607, 0.260682) (− 0.36526, 0.108517) (− 0.111607, 0.837064) (0.0101461, 0.09922946) (0.131899, 0.258376) | (− 0.200626, 0.790477) (− 0.192776, 0.788648) | 0 |
Figure 25f | (− 0.0943028, 1.42143) (0.022767, 0.256532) (− 0.0923406, 1.41911) (0.0243002, 0.219746) (− 0.102061, 1.39976) (0.0145801, 0.200401) | (− 0.185354, 0.385965) (− 0.189541, 0.316468) (− 0.199261, 0.316468) | 0 |
Figure | PFP | CFP | \(\sigma_{2}\) |
---|---|---|---|
Figure 26a | (− 0.295838, 0.1078713) (− 0.105413, 0.755748) (1.29557, 0.0492287) (− 0.39105, 0.108455) (− 0.187024, 1.36209) (− 0.369843, 0.121979) (− 0.179832, 1.44755) (− 0.132329, 0.27164) | (− 0.105413, 0.755748) (− 0.202922, 1.36418) (− 0.179832, 1.44755) | − 0.0514914 |
Figure 26b | (− 0.88361, 0.0390682) (− 0.0943028, 0.692148) (0.0227627, 0.111633) (− 0.296462, 0.136059) (− 0.189541, − 0.39118) (− 0.0923406, 1.22946) (0.00486003, − 0.488743) (− 0.289412, 0.02978291) (− 0.198361, 2.76917) (0.00975546, -0.518095) | (− 0.198361, 0.0390682) (− 0.189541, -0.39118) (− 0.198361, 2.76917) | − 0.0514914 |
Figure 26c | (− 0.39105, 0.108455) (− 0.187024, 1.36209) (− 0.385552, 0.104751) (− 0.92776, 1.36418) | (− 0.202922, 1.36418) (− 0.192776, 1.36418) | − 0.0514914 |
Figure 26d | (− 0.296462, 0.136059) (− 0.189541, − 0.39118) (− 0.0923406, 1.22946) (0.00486003, − 0.488743) (− 0.296462, 0.136059) (0.189541, − 0.410643) (− 0.09234061, 1.22946) (0.00486003, − 0.488743) | (− 0.189541, − 0.39118) | − 0.0514914 |
Figure 26e | (− 0.39105, 0.108455) (− 0.187024, 1.36209) (− 0.395698, 0.104751) (− 0.202922, 1.36795) (− 0.385552, 0.108521) (− 0.213068, 1.36041) | (− 0.202922, 1.36418) (− 0.202922, 1.36795) (− 0.213068, 1.36041) | − 0.0514914 |
Figure 26f | (− 0.296462, 0.136059) (− 0.189541, − 0.39118) (− 0.0923406, 1.22946) (0.00486003, − 0.488743) (− 0.296462, 0.116534) (− 0.17982, 0.371592) (− 0.0923406, 1.22946) (0.00486003, − 0.488743) (− 0.296462, 0.116536) (− 0.199261, − 0.371592) (− 0.102061, 1.24899) (0.00486003, − 0.449693) | (− 0.189541, − 0.39118) (− 0.17982, − 0.371592) (− 0.199261, − 0.371592) | − 0.0514914 |