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On the Zero-Hopf Bifurcation of the Lotka–Volterra Systems in
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## Abstract

**:**

## 1. Introduction and Statement of Results

**Theorem**

**1.**

## 2. The Algorithm for Computing the Periodic Solutions Bifurcating from a Zero-Hopf Equilibrium

**(1)**- Since we want to apply the averaging theory of order three (see the Appendix A) for studying the periodic solutions bifurcating from the zero–Hopf equilibrium at the origin and the averaging theory uses a small parameter $\epsilon $, we write the parameters of the system in the form$${\lambda}_{k}={\lambda}_{k0}+\epsilon {\lambda}_{k1}+{\epsilon}^{2}{\lambda}_{k2}+{\epsilon}^{3}{\lambda}_{k3},\phantom{\rule{2.em}{0ex}}\mathrm{for}k=1,\dots ,m.$$
**(2)**- Due to the fact that the zero–Hopf bifurcation will take place in a neighborhood of the origin, where it is localized the zero–Hopf equilibrium, we blow up this neighborhood doing the scaling of variables$$({x}_{1},\dots ,{x}_{n})=(\epsilon {X}_{1},\dots ,\epsilon {X}_{n}),$$$${\dot{X}}_{i}={F}_{i}({X}_{1},\dots ,{X}_{n},\lambda ,\epsilon )\phantom{\rule{2.em}{0ex}}\mathrm{for}i=1,\dots ,n.$$
**(3)**- In order to simplify the future computations and also for applying the averaging theory described in the Appendix A we need that the right hand part of the differential system starts with order $\epsilon $, for these two reasons we shall pass the linear part of the differential system (3) to its real Jordan normal form doing a convenient linear change of variables $({X}_{1},\dots ,{X}_{n})\to ({u}_{1},\dots ,{u}_{n})$. Thus the differential system (3) in the new variables $({u}_{1},\dots ,{u}_{n})$ writes$$\begin{array}{cc}\hfill {\dot{u}}_{1}=& -\omega {u}_{2}+\epsilon {g}_{1}({u}_{1},\dots ,{u}_{n},\lambda ,\epsilon ),\phantom{\rule{5.69046pt}{0ex}}\hfill \\ \hfill {\dot{u}}_{2}=& \omega {u}_{1}+\epsilon {g}_{2}({u}_{1},\dots ,{u}_{n},\lambda ,\epsilon ),\phantom{\rule{5.69046pt}{0ex}}\hfill \\ \hfill {\dot{u}}_{3}=& \epsilon {g}_{3}({u}_{1},\dots ,{u}_{n},\lambda ,\epsilon ),\phantom{\rule{5.69046pt}{0ex}}\hfill \\ \hfill \cdots \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}& \phantom{\rule{2.em}{0ex}}\cdots \phantom{\rule{5.69046pt}{0ex}}\hfill \\ \hfill {\dot{u}}_{n}=& \epsilon {g}_{n}({u}_{1},\dots ,{u}_{n},\lambda ,\epsilon ).\hfill \end{array}$$
**(4)**- In order to apply the averaging theory to a differential system the right hand part of that differential system must be a periodic function in the independent variable of the system, see again the Appendix A. For this reason we first pass the differential system (4) to the generalized cylindrical coordinates $(r,\theta ,{u}_{3},\dots ,{u}_{n})$ where ${u}_{1}=rcos\theta $ and ${u}_{2}=rsin\theta $, and system (4) becomes$$\begin{array}{cc}\hfill \dot{r}=& \epsilon {G}_{1}(r,\theta ,{u}_{3},\dots ,{u}_{n},\lambda ,\epsilon ),\phantom{\rule{5.69046pt}{0ex}}\hfill \\ \hfill \dot{\theta}=& \omega +\epsilon {G}_{2}(r,\theta ,{u}_{3},\dots ,{u}_{n},\lambda ,\epsilon ),\phantom{\rule{5.69046pt}{0ex}}\hfill \\ \hfill {\dot{u}}_{3}=& \epsilon {G}_{3}(r,\theta ,{u}_{3},\dots ,{u}_{n},\lambda ,\epsilon ),\phantom{\rule{5.69046pt}{0ex}}\hfill \\ \hfill \cdots \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}& \phantom{\rule{2.em}{0ex}}\cdots \phantom{\rule{5.69046pt}{0ex}}\hfill \\ \hfill {\dot{u}}_{n}=& \epsilon {G}_{n}(r,\theta ,{u}_{3},\dots ,{u}_{n},\lambda ,\epsilon ).\hfill \end{array}$$Now this differential system has its right hand part periodic in the variable $\theta $, because this variable appears only through the functions $cos\theta $ and $sin\theta $. Since the cylindrical coordinates are not well defined at $r=0$, we are studying only the periodic orbits which does not intersect the set $r=0$.
**(5)**- Now we take the variable $\theta $ as the new independent variable, and system (5) in this new independent variable writes$$\begin{array}{cc}\hfill {r}^{\prime}=& \epsilon {H}_{1}(r,\theta ,{u}_{3},\dots ,{u}_{n},\lambda ,\epsilon ),\phantom{\rule{5.69046pt}{0ex}}\hfill \\ \hfill {u}_{3}^{\prime}=& \epsilon {H}_{3}(r,\theta ,{u}_{3},\dots ,{u}_{n},\lambda ,\epsilon ),\phantom{\rule{5.69046pt}{0ex}}\hfill \\ \hfill \cdots \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}& \phantom{\rule{2.em}{0ex}}\cdots \phantom{\rule{5.69046pt}{0ex}}\hfill \\ \hfill {u}_{n}^{\prime}=& \epsilon {H}_{n}(r,\theta ,{u}_{3},\dots ,{u}_{n},\lambda ,\epsilon ),\hfill \end{array}$$
**(6)**- We apply the averaging theory of third order and according with it we may get s periodic solutions $({r}^{k}(\theta ,\epsilon ),{u}_{3}^{k}(\theta ,\epsilon ),\dots ,{u}_{n}^{k}(\theta ,\epsilon ))$ of system (6) for $k=1,\dots ,s$ such that$$({r}^{k}(0,\epsilon ),{u}_{3}^{k}(0,\epsilon ),\dots ,{u}_{n}^{k}(0,\epsilon ))=({r}^{k*},{u}_{3}^{k*},\dots ,{u}_{n}^{k*})+O\left(\epsilon \right),$$
**(7)**- Now we go back through the changes of variables until the initial differential system (2), and we shall see how look the periodic solutions $({r}^{k}(\theta ,\epsilon ),{u}_{3}^{k}(\theta ,\epsilon ),\dots $, ${u}_{n}^{k}(\theta ,\epsilon ))$ in the initial differential system. First the periodic solutions (7) in the differential system (5) are $({r}^{k}(t,\epsilon ),{\theta}^{k}(t,\epsilon ),{u}_{3}^{k}(t,\epsilon ),\dots ,{u}_{n}^{k}(t,\epsilon ))$ verifying$$({r}^{k}(0,\epsilon ),{\theta}^{k}(0,\epsilon ),{u}_{3}^{k}(0,\epsilon ),\dots ,{u}_{n}^{k}(t,\epsilon ))=({r}^{k*},0,{u}_{3}^{k*},\dots ,{u}_{n}^{k*})+O\left(\epsilon \right),$$

## 3. Proof of Theorem 1

- Case 1:
- ${a}_{220}=-{a}_{230}$,${a}_{331}=({a}_{121}{a}_{230}{a}_{310}+{a}_{131}{a}_{230}{a}_{310}-{a}_{221}{\omega}^{2})/{\omega}^{2}$,${a}_{111}=(-{a}_{121}{a}_{230}{a}_{310}-{a}_{131}{a}_{230}{a}_{310})/{\omega}^{2}$.
- Case 2:
- ${a}_{310}=0$, ${a}_{331}=-{a}_{221}$, ${a}_{111}=0$.
- Case 3:
- ${a}_{220}=0$,${a}_{331}=({a}_{121}{a}_{230}{a}_{310}-{a}_{221}{\omega}^{2})/{\omega}^{2}$,${a}_{111}=-\left({a}_{121}{a}_{230}{a}_{310}\right)/{\omega}^{2}$.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Averaging Theory of First, Second and Third Order

- (i)
- ${F}_{1}(t,\xb7)\in {C}^{2}\left(D\right)$, ${F}_{2}(t,\xb7)\in {C}^{1}\left(D\right)$ for all $t\in \mathbb{R}$, ${F}_{1}$, ${F}_{2}$, ${F}_{3}$, R, ${D}_{x}^{2}{F}_{1}$, ${D}_{x}{F}_{2}$ are locally Lipschitz with respect to x, and R is twice differentiable with respect to $\epsilon $.We define ${F}_{k0}:D\to \mathbb{R}$ for $k=1,2,3$ as$$\begin{array}{cc}\hfill {f}_{1}\left(x\right)=& {\displaystyle {\displaystyle \frac{1}{T}}{\int}_{0}^{T}{F}_{1}(s,x)ds,\phantom{\rule{5.69046pt}{0ex}}}\hfill \\ \hfill {f}_{2}\left(x\right)=& {\displaystyle {\displaystyle \frac{1}{T}}{\int}_{0}^{T}\left[{D}_{x}{F}_{1}(s,x)\xb7{y}_{1}(s,x)+{F}_{2}(s,x)\right]ds,\phantom{\rule{5.69046pt}{0ex}}}\hfill \\ \hfill {f}_{3}\left(x\right)=& {\displaystyle {\displaystyle \frac{1}{T}}{\int}_{0}^{T}[{\displaystyle \frac{1}{2}}{y}_{1}{(s,x)}^{T}{\displaystyle \frac{{\partial}^{2}{F}_{1}}{\partial {x}^{2}}}(s,x){y}_{1}(s,x)+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\partial {F}_{1}}{\partial x}}(s,x){y}_{2}(s,x)}\hfill \\ & {\displaystyle +{\displaystyle \frac{\partial {F}_{2}}{\partial x}}(s,x){y}_{1}(s,x)+{F}_{3}(s,x)]ds,}\hfill \end{array}$$$$\begin{array}{c}{\displaystyle {y}_{1}(s,x)={\int}_{0}^{s}{F}_{1}(t,x)dt,}\hfill \\ {\displaystyle {y}_{2}(s,x)={\int}_{0}^{s}\left[{\displaystyle \frac{\partial {F}_{1}}{\partial x}}(t,x){\int}_{0}^{t}{F}_{1}(r,x)dr+{F}_{2}(t,x)\right]dt.}\hfill \end{array}$$
- (ii)
- For an open and bounded set $V\subset D$ and for each $\epsilon \in (-{\epsilon}_{f},{\epsilon}_{f})\backslash \left\{0\right\}$, there exists $a\in V$ such that ${f}_{1}\left(a\right)+\epsilon {f}_{2}\left(a\right)+{\epsilon}^{2}{f}_{3}\left(a\right)=0$ and ${d}_{B}({f}_{1}+\epsilon {f}_{2}+{\epsilon}^{2}{f}_{3},V,{a}_{\epsilon})\ne 0$ (i.e., the Brouwer degree of the function ${f}_{1}+\epsilon {f}_{2}+{\epsilon}^{2}{f}_{3}$ at the point a is not zero).

## References

- Baldomá, I.; Castejón, O.; Seara, M.T. Exponentially small heteroclinic breakdown in the generic Hopf-Zero singularity. J. Dyn. Diff. Equ.
**2013**, 25, 335–392. [Google Scholar] [CrossRef] [Green Version] - Broer, H.W.; Vegter, G. Subordinate Silnikov bifurcations near some singularities of vector fields having low codimension. Ergodic Theory Dyn. Syst.
**1984**, 4, 509–525. [Google Scholar] [CrossRef] [Green Version] - Champneys, A.R.; Kirk, V. The entwined wiggling of homoclinic curves emerging from saddle-node/Hopf instabilities. Physica D
**2004**, 195, 77–105. [Google Scholar] [CrossRef] [Green Version] - Dumortier, F.; Ibá nez, S.; Kokubu, H.; Simó, C. About the unfolding of a Hopf-zero singularity. Discrete Contin. Dyn. Syst.
**2013**, 33, 4435–4471. [Google Scholar] [CrossRef] - Guckenheimer, J. On a codimension two bifurcation. Lect. Notes Math.
**1980**, 898, 99–142. [Google Scholar] - Guckenheimer, J.; Holmes, P. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields; Springer: Berlin/Heidelberg, Germany, 1983. [Google Scholar]
- Kuznetsov, Y.A. Elements of Applied Bifurcation Theory, 3rd ed.; Springer: Berlin/Heidelberg, Germany, 2004. [Google Scholar]
- Scheurle, J.; Marsden, J. Bifurcation to quasi-periodic tori in the interaction of steady state and Hopf bifurcations. SIAM J. Math. Anal.
**1984**, 15, 1055–1074. [Google Scholar] [CrossRef] [Green Version] - Hirsch, M.W. Systems of differential equations that are competitive or cooperative. I: Limit sets. SIAM J. Math. Anal.
**1982**, 13, 167–169. [Google Scholar] [CrossRef] - Hirsch, M.W. Systems of differential equations that are competitive or cooperative. II: Convergence almost everywhere. SIAM J. Math. Anal.
**1985**, 16, 423–439. [Google Scholar] [CrossRef] - Hirsch, M.W. Stability and convergence in stronly monotone dynamical systems. J. Die Reine Angew. Mathematik
**1988**, 383, 1–53. [Google Scholar] - Hirsch, M.W. Systems of differential equations that are competitive or cooperative. III: Competing species. Nonlinearity
**1988**, 1, 117–124. [Google Scholar] [CrossRef] [Green Version] - Hirsch, M.W. Systems of differential equations that are competitive or cooperative. IV: Structural stability in 3-dimensional systems. SIAM J. Math. Anal.
**1990**, 21, 1225–1234. [Google Scholar] [CrossRef] - Hirsch, M.W. Systems of differential equations that are competitive or cooperative. V: Convergence in 3-dimensional systems. J. Differ. Equ.
**1989**, 80, 94–106. [Google Scholar] [CrossRef] [Green Version] - Hirsch, M.W. Systems of differential equations that are competitive or cooperative. VI: A local C
^{r}closing lemma for 3-dimensional systems. Ergod. Theory Dyn. Syst.**1990**, 11, 443–454. [Google Scholar] [CrossRef] - Lotka, A.J. Analytical note on certain rhythmic relations in organic systems. Proc. Natl. Acad. Sci. USA
**1920**, 6, 410–415. [Google Scholar] [CrossRef] [Green Version] - May, R.M. Stability and Complexity in Model Ecosystems; Princeton University Press: Princeton, NJ, USA, 1974. [Google Scholar]
- Smale, S. On the differential equations of species in competition. J. Math. Biol.
**1976**, 3, 5–7. [Google Scholar] [CrossRef] - Volterra, V. Lecons sur la Théorie Mathématique de la Lutte pour la vie; Gauthier Villars: Paris, France, 1931. [Google Scholar]
- Zeeman, M.L. Hopf bifurcations in competitive three-dimensional Lotka–Volterra systems. Dyn. Stab. Sys.
**1993**, 8, 189–217. [Google Scholar] [CrossRef] - Laval, G.; Pellat, R. Plasma Physics. Proceedings of Summer School of Theoretical Physics; Gordon and Breach: New York, NY, USA, 1975. [Google Scholar]
- Busse, F.H. Transition to Turbulence via the Statistical Limit Cycle Route Synergetics; Springer: Berlin/Heidelberg, Germany, 1978; p. 39. [Google Scholar]
- Llibre, J.; Xiao, D. Limit cycles bifurcating from a non-isolated zero-Hopf equilibrium of 3-dimensional differential systems. Proc. Am. Math. Soc.
**2014**, 142, 2047–2062. [Google Scholar] [CrossRef] [Green Version] - Arnold, V.I. Arnold’s Problems; Springer: Berlin/Heidelberg, Germany, 2000. [Google Scholar]
- Hofbauer, J.; So, J.W.H. Multiple limit cycles for three dimensional Lotka–Volterra equations. Appl. Math. Lett.
**1994**, 7, 65–70. [Google Scholar] [CrossRef] [Green Version] - Lu, Z.; Luo, Y. Two limit cycles in three-dimensional Lotka–Volterra systems. Comput. Math. Appl.
**2002**, 44, 51–66. [Google Scholar] [CrossRef] [Green Version] - Xiao, D.; Li, W. Limit cycles for the competitive three dimensional Lotka–Volterra system. Differ. Equ.
**2000**, 164, 1–15. [Google Scholar] [CrossRef] [Green Version] - Gyllenberg, M.; Yan, P. On the number of limit cycles for the three-dimensional Lotka–Volterra systems. Discrete Contin. Dyn. Syst. Ser. S
**2009**, 11, 347–352. [Google Scholar] [CrossRef] - Gyllenberg, M.; Yan, P.; Wang, Y. A 3D competitive LotkaVolterra system with three limit cycles: A falsification of a conjecture by Hofbauer and So. Appl. Math. Lett.
**2006**, 19, 1–7. [Google Scholar] [CrossRef] [Green Version] - Lu, Z.; Luo, Y. Three limit cycles for a three-dimensional Lotka–Volterra competitive system with a heteroclinic cycle. Comput. Math. Appl.
**2003**, 46, 231–238. [Google Scholar] [CrossRef] [Green Version] - Yu, P.; Han, M.; Xiao, D. Four small limit cycles around a Hopf singular point in 3-dimensional competitive Lotka–Volterra systems. J. Math. Anal. Appl.
**2016**, 436, 521–555. [Google Scholar] [CrossRef] - Giné, J.; Grau, M.; Llibre, J. Averaging theory at any order for computing periodic orbits. Physica D
**2013**, 250, 58–65. [Google Scholar] [CrossRef] [Green Version] - Giné, J.; Llibre, J.; Wu, K.S.; Zhang, X. Averaging methods of arbitrary order, periodic solutions and integrability. J. Differ. Equ.
**2016**, 260, 4130–4156. [Google Scholar] [CrossRef] [Green Version] - Llibre, J.; Novaes, D.D.; Teixeira, M.A. Higher order averaging theory for finding periodic solutions via Brouwer degree. Nonlinearity
**2014**, 27, 563–583. [Google Scholar] [CrossRef] [Green Version] - Buica, A.; Llibre, J. Averaging methods for finding periodic orbits via Brouwer degree. Bull. Sci. Math.
**2004**, 128, 7–22. [Google Scholar] [CrossRef] [Green Version] - Lloyd, N.G. Degree Theory; Cambridge University Press: Cambridge, UK, 1978. [Google Scholar]

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Han, M.; Llibre, J.; Tian, Y.
On the Zero-Hopf Bifurcation of the Lotka–Volterra Systems in *Mathematics* **2020**, *8*, 1137.
https://doi.org/10.3390/math8071137

**AMA Style**

Han M, Llibre J, Tian Y.
On the Zero-Hopf Bifurcation of the Lotka–Volterra Systems in *Mathematics*. 2020; 8(7):1137.
https://doi.org/10.3390/math8071137

**Chicago/Turabian Style**

Han, Maoan, Jaume Llibre, and Yun Tian.
2020. "On the Zero-Hopf Bifurcation of the Lotka–Volterra Systems in *Mathematics* 8, no. 7: 1137.
https://doi.org/10.3390/math8071137