Bifurcations and chaos in a two-dimensional discrete Hindmarsh–Rose model

Abstract

In this paper, the dynamics of a two-dimensional discrete Hindmarsh–Rose model is discussed. It is shown that the system undergoes flip bifurcation, Neimark–Sacker bifurcation, and 1:1 resonance by using a center manifold theorem and bifurcation theory. Furthermore, we present the numerical simulations not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, including orbits of period 3, 6, 15, cascades of period-doubling bifurcation in orbits of period 2, 4, 8, 16, quasiperiodic orbits, and chaotic sets. These results obtained in this paper show far richer dynamics of the discrete Hindmarsh–Rose model compared with the corresponding continuous model.

Appendix

In this section, we provide the proof of Theorem 3.4 and the expressions of bifurcation curves near the 1:1 resonance point.

Proof of Theorem 3.4

By Proposition 3.1 in [33] and corresponding results in [2, 17], we only need to check the nondegeneracy conditions

$$q_{20}(0) \neq0, \qquad p_{20}(0) + q_{11}(0)- q_{20}(0)\neq0, $$

and the transversality condition

$$\mathop{\mathrm{det}} D_{\mu}\rho(0)\neq0. $$

It is easy to see that the above conditions hold if b>2d and δ≠δ ₃.

Furthermore, the invariant circle created at the Neimark–Sacker bifurcation is stable (resp. unstable) if

$$q_{20}(0) \bigl(p_{20}(0) + q_{11}(0)- q_{20}(0) \bigr) < 0\quad (\mbox{resp}. > 0). $$

It is easy to calculate that the invariant circle is stable (resp., unstable) when $\delta>(\mbox{resp}.<)\frac{2d-b}{d-b}$.

Next, we give the lower order terms of μ to express the bifurcation curves near 1:1 resonance point by Maple.

(i) The fold bifurcation curve can be expressed as

$$\begin{aligned} & {F_{10}\mu_{1}+F_{01}\mu_{2}+ \frac{1}{2}F_{20}\mu_{1}^{2}+F_{11} \mu_{1}\mu_{2}+\frac{1}{2}F_{02} \mu_{2}^{2} }\\ & {\quad {}=O \bigl(\bigl(|\mu_{1}|+| \mu_{2}|\bigr)^{3} \bigr), } \end{aligned}$$

where

$$\begin{aligned} & {F_{10}=\displaystyle{\frac{2 ( 2 d-b-\delta d+\delta b ) ^{4}}{ ( b-d ) ^{3}{d}^{3}}}, }\\ & {F_{01}=\displaystyle{\frac{ 16 ( 2 d-b-\delta d+\delta b ) ^{4}}{ (b-d ) ^{3}}}, } \end{aligned}$$

$$\begin{aligned} F_{20} =& \frac{ ( 2 d-b-\delta d+\delta b ) ^{ 2}}{72 (b-d ) ^{4}{d}^{6}} \bigl( 972 d{\delta}^{4}{b}^{3}+1632 {\delta}^{4}{d}^{3}b-1870 { \delta}^{4}{d}^{2}{b}^{2}-6 {b}^{3}d{\delta}^{3}-840 {b}^{3}d{\delta }^{2} \\ &{}-986 {\delta}^{5}{d}^{3}b+1278 {\delta}^{5}{d}^{2}{b}^{2}-2808 {d}^{3}\delta b+432 {d}^{2}\delta {b}^{2}-3084 {d}^{3}{\delta}^{2}b+2640 {d}^{2}{\delta}^{2}{b}^{2} \\ &{}+1320 {d}^{3}{\delta}^{3}b-948 { d}^{2}{ \delta}^{3}{b}^{2}+432 {b}^{3}d\delta+3888 {d}^{3}b+1728 {d} ^{4}\delta+1296 {d}^{4}{ \delta}^{2}-492 {d}^{4}{\delta}^{3} \\ & {}-1296 {b }^{2}{d}^{2}-532 {\delta}^{4}{d}^{4}-30 {b}^{4}{\delta}^{2}+126 {b} ^{4}{ \delta}^{3}-202 {\delta}^{4}{b}^{4}+280 { \delta}^{5}{d}^{4}+146 {\delta}^{5}{b}^{4}-40 {\delta}^{6}{d}^{4} \\ &{}-40 {\delta}^{6}{b}^{4}+ 2592 {d}^{5} \delta-4752 {d}^{5}{\delta}^{2}+1728 {d}^{5}{ \delta}^{3 }+912 {\delta}^{4}{d}^{5}-36 {b}^{5}{\delta}^{3}+120 {\delta}^{4}{b }^{5}-424 {\delta}^{5}{d}^{5} \\ &{}-145 {\delta}^{5}{b}^{5}-48 {\delta}^{ 6}{d}^{5}+75 {\delta}^{6}{b}^{5}+18 {\delta}^{7}{d}^{5}-15 {\delta} ^{7}{b}^{5}-{\delta}^{8}{d}^{5}+{ \delta}^{8}{b}^{5}-3240 {d}^{4}-718 d{ \delta}^{5}{b}^{3} \\ &{}+160 {\delta}^{6}{d}^{3}b-240 {\delta}^{6}{d}^ {2}{b}^{2}+160 d{\delta}^{6}{b}^{3}-6480 {d}^{4}\delta b+5184 {d}^ {3}\delta {b}^{2}-1296 {d}^{2}\delta {b}^{3} \\ &{}+11664 {d}^{4}{\delta} ^{2}b -8640 {d}^{3}{ \delta}^{2}{b}^{2}+1296 {d}^{2}{ \delta}^{2}{b}^{3 }-3168 {d}^{4}{ \delta}^{3}b+2556 {d}^{2}{\delta}^{3}{b}^{3}+432 {b} ^{4}d{\delta}^{2} \\ &{}-1080 {b}^{4}d{\delta}^{3}-3696 {\delta}^{4}{d}^{4}b+5556 {\delta }^{4}{d}^{3}{b}^{2} -3552 { \delta}^{4}{d}^{2}{b}^{3}+ 660 d{ \delta}^{4}{b}^{4}+1448 {\delta}^{5}{d}^{4}b \\ &{}-1654 {\delta}^{5 }{d}^{3}{b}^{2}+515 { \delta}^{5}{d}^{2}{b}^{3}+260 d{ \delta}^{5}{b}^ {4} +264 {\delta}^{6}{d}^{4}b-579 {\delta}^{6}{d}^{3}{b}^{2}+633 { \delta}^{6}{b}^{3}{d}^{2}-345 { \delta}^{6}{b}^{4}d \\ &{}-87 {\delta}^{7}{d }^{4}b+168 {\delta}^{7}{d}^{3}{b}^{2}-162 {\delta }^{7}{b}^{3}{d}^{2}+78 { \delta}^{7}{b}^{4}d+5 {\delta }^{8}{d}^{4}b-10 {\delta}^{8}{d}^ {3}{b}^{2}+10 { \delta}^{8}{b}^{3}{d}^{2}-5 {\delta}^{8}{b}^{4}d \bigr), \\ F_{11} =& {\frac{\delta ( 2 d-b-\delta d+\delta b ) ^{3}}{ 18 (b-d ) ^{4}{d}^{3}}} \bigl( -216 {d}^{3}-30 d{ \delta }^{4}{b}^{3}+12 {\delta}^{4}{d}^{3} b+13 {\delta}^{4}{d}^{2}{b}^{2}-36 {b}^{3}d{\delta}^{3}+36 {b}^{3}d { \delta}^{2} \\ &{}+38 {\delta}^{5}{d}^{3}b-54 {\delta}^{5}{d}^{2}{b}^{2}- 156 {d}^{2}\delta b-396 {d}^{2}{\delta}^{2}b+488 {d}^{2}{\delta}^{ 3}b+12 {b}^{2}d\delta+270 {b}^{2}d{\delta}^{2}-394 {b}^{2}d{\delta} ^{3} \\ &{}-120 {\delta}^{4}{d}^{2}b+120 d{\delta}^{4}{b}^{2}+180 {d}^{3}{ \delta}^{2}b-144 {d}^{2}{ \delta}^{2}{b}^{2}-216 {d}^{3}{ \delta}^{3}b +174 {d}^{2}{\delta}^{3}{b}^{2}+114 {d}^{3}\delta \\ &{}+198 {d}^{3}{ \delta}^{2}-194 {d}^{3}{ \delta}^{3}+108 b{d}^{2}+40 {\delta}^{4}{d} ^{3}+12 {b}^{3}\delta-72 {b}^{3}{ \delta}^{2}+100 {b}^{3}{\delta}^{3 }-40 { \delta}^{4}{b}^{3} \\ &{}-72 {d}^{4}{\delta}^{2}+84 {d}^{4}{ \delta}^ {3}-8 {\delta}^{4}{d}^{4}-6 {b}^{4}{\delta}^{3}+13 {\delta}^{4}{b}^ {4}-10 {\delta}^{5}{d}^{4}-8 {\delta}^{5}{b}^{4}+{ \delta}^{6}{d}^{4} +{\delta}^{6}{b}^{4}+34 d{\delta}^{5}{b}^{3} \\ &{}-4 {\delta}^{6}{d}^{3}b+ 6 {\delta}^{6}{d}^{2}{b}^{2}-4 d{\delta}^{6}{b}^{3} \bigr), \\ F_{02} =&{ \frac{8{\delta}^{2} ( 2 d-b-\delta d+\delta b ) ^{4}}{9(b-d) ^{4}}} \bigl( -72 {d}^{2}-162 d\delta b+80 d{ \delta}^{2}b+ 108 \delta {d}^{2}-40 {\delta}^{2}{d}^{2}+72 bd+54 \delta {b}^{2 } \\ &{}-40 {\delta}^{2}{b}^{2}-18 {b}^{2}+2 { \delta}^{3}{d}^{3}-5 { \delta}^{3}b{d}^{2}-{ \delta}^{3}{b}^{3}-{\delta}^{4}{d}^{3}+{ \delta}^{ 4}{b}^{3}+4 {\delta}^{3}d{b}^{2}+3 {\delta}^{4}{d}^{2}b-3 {\delta}^ {4}{b}^{2}d \bigr). \end{aligned}$$

 (ii) The Neimark–Sacker bifurcation curve can be expressed as

$$H_{10}\mu_{1}+H_{01}\mu_{2}+ \frac{1}{2}H_{20}\mu_{1}^{2}+H_{11 } \mu_{1}\mu_{2}+\frac{1}{2}H_{02} \mu_{2}^{2}=O \bigl(\bigl(|\mu_{1}|+| \mu_{2}|\bigr)^{3} \bigr) $$

and

$$\mathit{HF}_{10}\mu_{1}+\mathit{HF}_{01}\mu_{2}+O \bigl(\bigl(|\mu_{1}|+|\mu_{2}|\bigr)^{2} \bigr)<0, $$

where

$$\begin{aligned} H_{10} =&\displaystyle{\frac{2 ( 2 d-b-\delta d+\delta b ) ^{4}}{ (b-d ) ^{3}{d}^{3}}}, \\ H_{01} =&\displaystyle{\frac{16 ( 2 d-b-\delta d+\delta b ) ^{4}}{ (b - d ) ^{3}}}, \\ H_{20} =&{\frac{ ( 2 d-b-\delta d+\delta b ) ^{ 3}}{72 (b -d ) ^{3}{d}^{6}}} \bigl( 1296 {d}^{2}-432 d \delta b+588 d{\delta}^{2}b-108 { \delta}^{3}db-648 \delta {d}^{2}-612 {\delta}^{2}{d}^{2} \\ &{}+12 { \delta}^{2}{b}^{2}+132 {\delta}^{3}{d}^{2}-30 {\delta}^{3}{b}^{2}+ 1296 {d}^{2}\delta b-1296 {d}^{2}{\delta}^{2}b-792 {d}^{2}{\delta} ^{3}b-432 {b}^{2}d{\delta}^{2} \\ &{}+756 {b}^{2}d{\delta}^{3}+768 {\delta }^{4}{d}^{2}b-228 d{\delta}^{4}{b}^{2}+92 {\delta}^{5}{d}^{2}b-137 d{\delta}^{5}{b}^{2}-1296 {d}^{3}\delta+1728 {d}^{3}{\delta}^{2} \\ &{}-456 {\delta}^{4}{d}^{3}+36 {b}^{3}{ \delta}^{3}-84 {\delta}^{4}{b}^{3}- 16 { \delta}^{5}{d}^{3}+61 {\delta}^{5}{b}^{3}+40 {\delta}^{4}{d}^{2 }+26 {\delta}^{4}{b}^{2}+16 {\delta}^{6}{d}^{3} \\ &{}-14 {\delta}^{6}{b}^ {3}-8 {\delta}^{5}{d}^{2}-8 {\delta}^{5}{b}^{2}-{\delta}^{7}{d}^{3}+ {\delta}^{7}{b}^{3}-66 {\delta}^{4}db-46 { \delta}^{6}{d}^{2}b+44 { \delta}^{6}{b}^{2}d \\ &{}+16 {\delta}^{5}db-3 {\delta}^{7}d{b}^{2}+3 { \delta}^{7}{d}^{2}b \bigr) , \\ H_{11} =&{ \frac{{\delta}^{2} ( 2 d-b-\delta d+\delta b ) ^ {3}}{18 (b -d ) ^{3}{d}^{3}}} \bigl( -114 {d}^{2}-66 d\delta b-54 d{ \delta}^{2}b+16 {\delta }^{3}db+66 \delta {d}^{2}+34 {\delta}^{2}{d}^{2} \\ &{}+114 bd-6 \delta {b}^{2}+20 {\delta}^{2}{b}^{2}-8 {\delta}^{3}{d}^{2}-8 {\delta}^{ 3}{b}^{2}-6 {b}^{2}-108 {d}^{2}\delta b+132 {d}^{2}{ \delta}^{2}b-4 {d}^{2}{\delta}^{3}b \\ &{}+36 {b}^{2}d\delta-42 {b}^{2}d{\delta}^{2}-17 {b}^{2}d{\delta}^{3}-28 {\delta}^{4}{d}^{2}b+26 d{\delta}^{4}{b}^{ 2}+3 {\delta}^{5}{d}^{2}b-3 d{\delta}^{5}{b}^{2}+72 {d}^{3}\delta \\ &{}-84 {d}^{3}{\delta}^{2}+8 {d}^{3}{ \delta}^{3}+10 {\delta}^{4}{d}^{3} -6 {b}^{3}{\delta}^{2}+13 {b}^{3}{ \delta}^{3}-8 {\delta}^{4}{b}^{3} -{ \delta}^{5}{d}^{3}+{\delta}^{5}{b}^{3} \bigr), \\ H_{02} =&\frac{8{\delta}^{3} ( 2 d-b-\delta d+\delta b ) ^{4}}{ 9 (b -d )^{3}} \bigl( -12 d+6 b+8 \delta d-8 \delta b-2 {\delta} ^{2}{d}^{2}+3 d{\delta}^{2}b \\ &{}-{\delta}^{2}{b}^{2}+{\delta}^{3}{d}^{2}- 2 {\delta}^{3}db+{\delta}^{3}{b}^{2} \bigr), \\ \mathit{HF}_{10} =&-{\frac{ ( 2 d-b-\delta d+\delta b )}{ 6 (b-d ) ^{2}{d}^{3}}} \bigl( 18 { d}^{2}-12 d \delta b-14 {\delta}^{2}{d}^{2}+21 d{\delta}^{2}b+3 \delta {b}^{2}-7 {\delta}^{2}{b}^{2} \\ &{}+4 {\delta}^{3}{d}^{2}-8 { \delta}^{3}db+4 { \delta}^{3}{b}^{2}+12 \delta {d}^{2} \bigr), \\ \mathit{HF}_{01} =&-{\frac{4\delta ( 2 d-b-\delta d+\delta b ) ^{2} ( 6 d-3 b-4 \delta d+4 \delta b ) }{3(b-d) ^{2}}}. \end{aligned}$$

 (iii) The homoclinic bifurcation curve can be expressed as

$$\mathit{Hom}_{10}\mu_{1}+\mathit{Hom}_{01}\mu_{2}+ \frac{1}{2}\mathit{Hom}_{20 }\mu_{1}^{2}+\mathit{Hom}_{11} \mu_{1}\mu_{2}+\frac{1}{2}\mathit{Hom}_{ 02} \mu_{2}^{2}=O \bigl(\bigl(|\mu_{1}|+| \mu_{2}|\bigr)^{3} \bigr) $$

and

$$\mathit{HF}_{10}\mu_{1}+\mathit{HF}_{01}\mu_{2}+O \bigl(\bigl(|\mu_{1}|+|\mu_{2}|\bigr)^{2} \bigr)<0, $$

where

$$\begin{aligned} \mathit{Hom}_{10} =&\displaystyle {\frac{2 ( 2 d-b-\delta d+\delta b ) ^{4}}{ (b -d ) ^{3}{d}^{3}}}, \\ \mathit{Hom}_{01} =&\displaystyle{\frac{ 16 ( 2 d-b-\delta d+\delta b )^{4}}{ (b-d ) ^{3}}}, \\ \mathit{Hom}_{20} =& {\frac{ ( 2 d-b-\delta d+\delta b ) ^{2}}{1800 (b-d ) ^{4}{d}^{6}}} \bigl( -18624 d{ \delta}^{4}{b}^{3}-45048 {\delta}^{4}{d}^{3}b+ 46252 {\delta}^{4}{d}^{2}{b}^{2}+28662 {b}^{3}d{\delta}^{3} \\ &{}-28056 { b}^{3}d{\delta}^{2}+13766 { \delta}^{5}{d}^{3}b-17442 {\delta}^{5}{d} ^{2}{b}^{2}-112536 {d}^{3}\delta b+21384 {d}^{2}\delta {b}^{2} \\ &{}-31236 {d}^{3}{\delta}^{2}b+62472 {d}^{2}{ \delta}^{2}{b}^{2}+87096 { d}^{3}{ \delta}^{3}b-83676 {d}^{2}{\delta}^{3}{b}^{2}+10800 {b}^{3}d \delta \\ &{}+97200 {d}^{3}b+85536 {d}^{4}\delta-2880 {d}^{4}{\delta}^{2}- 31116 {d}^{4}{ \delta}^{3}-32400 {b}^{2}{d}^{2}+15316 { \delta}^{4}{d }^{4} \\ &{}+132 {b}^{4}{\delta}^{2}-966 {b}^{4}{ \delta}^{3}+2104 {\delta} ^{4}{b}^{4}-3976 { \delta}^{5}{d}^{4}-1838 {\delta}^{5}{b}^{4}+568 { \delta}^{6}{d}^{4}+568 {\delta}^{6}{b}^{4} \\ &{}+64800 {d}^{5}\delta- 118800 {d}^{5}{ \delta}^{2}+43200 {d}^{5}{\delta}^{3}+22800 { \delta} ^{4}{d}^{5}-900 {b}^{5}{ \delta}^{3}+3000 {\delta}^{4}{b}^{5} \\ &{}-10600 {\delta}^{5}{d}^{5}-3625 { \delta}^{5}{b}^{5}-1200 {\delta}^{6}{d}^{5 }+1875 {\delta}^{6}{b}^{5}+450 {\delta}^{7}{d}^{5}-375 {\delta}^{7} {b}^{5}-25 {\delta}^{8}{d}^{5} \\ &{}+25 {\delta}^{8}{b}^{5}-49248 {d}^{4} +9490 d{\delta}^{5}{b}^{3}-2272 {\delta}^{6}{d}^{3}b+3408 {\delta}^ {6}{d}^{2}{b}^{2}-2272 d{ \delta}^{6}{b}^{3} \\ &{}-162000 {d}^{4}\delta b+ 129600 {d}^{3}\delta {b}^{2}-32400 {d}^{2}\delta {b}^{3}+291600 { d}^{4}{\delta}^{2}b-216000 {d}^{3}{ \delta}^{2}{b}^{2} \\ &{}+32400 {d}^{2}{ \delta}^{2}{b}^{3}-79200 {d}^{4}{\delta}^{3}b+63900 {d}^{2}{ \delta}^ {3}{b}^{3}+10800 {b}^{4}d{ \delta}^{2}-27000 {b}^{4}d{\delta}^{3} \\ &{}-92400 {\delta}^{4}{d}^{4}b+138900 {\delta }^{4}{d}^{3}{b}^{2}-88800 {\delta}^{4}{d}^{2}{b}^{3}+16500 d{\delta}^{4}{b}^{4}+36200 {\delta} ^{5}{d}^{4}b \\ &{}-41350 {\delta}^{5}{d}^{3}{b}^{2}+12875 { \delta}^{5}{d}^ {2}{b}^{3}+6500 d{ \delta}^{5}{b}^{4}+6600 {\delta}^{6}{d}^{4}b-14475 {\delta}^{6}{d}^{3}{b}^{2} \\ &{}+15825 {\delta}^{6}{b}^{3}{d}^{2}-8625 { \delta}^{6}{b}^{4}d-2175 {\delta}^{7}{d}^{4}b+4200 {\delta}^{7}{d}^{ 3}{b}^{2}-4050 { \delta}^{7}{b}^{3}{d}^{2} \\ &{}+1950 {\delta}^{7}{b}^{4}d+ 125 {\delta}^{8}{d}^{4}b-250 {\delta}^{8}{d}^{3}{b}^{2}+250 {\delta }^{8}{b}^{3}{d}^{2}-125 {\delta}^{8}{b}^{4}d \bigr), \\ \mathit{Hom}_{11} =&{ \frac{\delta ( 2 d-b-\delta d+\delta b ) ^{3}}{450 (b-d ) ^{4}{d}^{3}}} \bigl( 5184 {d}^{3}-750 d{\delta}^{4}{b}^{3}+300 { \delta}^{4}{d}^{3}b+325 {\delta}^{4}{d}^{2}{b}^{2} \\ &{}-900 {b}^{3}d{ \delta}^{3}+900 {b}^{3}d{ \delta}^{2}+950 {\delta}^{5}{d}^{3}b-1350 { \delta}^{5}{d}^{2}{b}^{2}-7428 {d}^{2} \delta b+15972 {d}^{2}{ \delta}^{2}b \\ &{}-7400 {d}^{2}{\delta}^{3}b+5592 {b}^{2}d \delta-9420 {b} ^{2}d{\delta}^{2}+5830 {b}^{2}d{ \delta}^{3}+1704 {\delta}^{4}{d}^{2} b-1704 d{ \delta}^{4}{b}^{2} \\ &{}+4500 {d}^{3}{\delta}^{2}b-3600 {d}^{2}{ \delta}^{2}{b}^{2}-5400 {d}^{3}{ \delta}^{3}b+4350 {d}^{2}{\delta}^{3 }{b}^{2}+2850 {d}^{3}\delta-7986 {d}^{3}{\delta}^{2} \\ &{}+2990 {d}^{3}{ \delta}^{3}-2592 b{d}^{2}-568 {\delta}^{4}{d}^{3}-582 {b}^{3}\delta +1434 {b}^{3}{\delta}^{2}-1420 {b}^{3}{ \delta}^{3}+568 {\delta}^{4} {b}^{3} \\ &{}-1800 {d}^{4}{\delta}^{2}+2100 {d}^{4}{ \delta}^{3}-200 { \delta}^{4}{d}^{4}-150 {b}^{4}{\delta}^{3}+325 {\delta}^{4}{b}^{4}- 250 {\delta}^{5}{d}^{4}-200 {\delta}^{5}{b}^{4} \\ &{}+25 {\delta}^{6}{d}^ {4}+25 {\delta}^{6}{b}^{4}+850 d{\delta}^{5}{b}^{3}-100 {\delta}^{6 }{d}^{3}b+150 {\delta}^{6}{d}^{2}{b}^{2}-100 d{ \delta}^{6}{b}^{3} \bigr), \\ \mathit{Hom}_{02} =&{\frac{8{\delta}^{2} ( 2 d-b-\delta d+\delta b ) ^{4}}{225 (b-d ) ^{4}}} \bigl(1728 {d}^{2}+3006 d \delta b-1136 d{\delta} ^{2}b-2004 \delta {d}^{2}+568 { \delta}^{2}{d}^{2} \\ &{}-1728 bd-1002 \delta {b}^{2}+568 {\delta}^{2}{b}^{2}+432 {b}^{2}-125 {d}^{2}{ \delta}^{3}b+100 {b}^{2}d{\delta}^{3}+75 {\delta}^{4}{d}^{2}b \\ &{}-75d\delta^{4}{b}^{2}+50d^{3}\delta^{3}-25\delta^{4}d^{3}-25 b^{3}\delta^{3}+25\delta^{4}b^{3} \bigr), \end{aligned}$$

and HF ₁₀ and HF ₀₁ are given in (ii).  □