Abstract. We analyze the dynamics of a class of Z6−equivariant sys-tems of the form z ̇ = pz2z ̄ ... more Abstract. We analyze the dynamics of a class of Z6−equivariant sys-tems of the form z ̇ = pz2z ̄ + sz3z̄2 − z̄5, where z is complex, the time t is real, while p and s are complex parameters. This study is the natural continuation of a previous work (M.J. Álvarez, A. Gasull, R. Prohens, Proc. Am. Math. Soc. 136, (2008), 1035–1043) on the normal form of Z4−equivariant systems. Our study uses the reduction of the equation to an Abel one, and provide criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle surrounding either 1, 7 or 13 critical points, the origin being always one of these points. Keywords: Planar autonomous ordinary differential equations, symmetric polinomial systems, limit cycles AMS Subject Classifications:
We analyze the dynamics of a class of Z2n-equivariant dif- ferential equations of the form u z = ... more We analyze the dynamics of a class of Z2n-equivariant dif- ferential equations of the form u z = pz n 1 ¯ z n 2 +sz nn 1 ¯ 2n 1 , where z is complex, the time t is real, while p and s are complex parameters. This study is the generalisation to Z2n of previous works with Z4 and Z6 symmetry. We reduce the problem of finding limit cycles to an Abel equation, and provide criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle that surrounds either 1, 2n+1 or 4n+1 equilibria, the origin being always one of these points.
We analyze the generating mechanisms for heteroclinic cycles in Z_2×Z_2×Z_2--equivariant ODEs, no... more We analyze the generating mechanisms for heteroclinic cycles in Z_2×Z_2×Z_2--equivariant ODEs, not involving Hopf bifurcations. Such cycles have been observed in particle physics systems with the mentioned symmetry, in absence of the Hopf bifurcation, see bury and Park, and as far as we know, there is no available theoretical data explaining these phenomena. We use singularity theory to study the equivalence in the group-symmetric context, as well as the recognition problem for the simplest bifurcation problems with this symmetry group. Singularity results highlight different mechanisms for the appearance of heteroclinic cycles, based on the transition between the bifurcating branches. On the other hand, we analyze the heteroclinic cycle of a generic dynamical system with the symmetry of the group Z_2×Z_2×Z_2 acting on a eight--dimensional torus T^8, constructed via a Cayley graph, under weak coupling. We identify the conditions for heteroclinic cycle between four equilibria in the ...
The existence and spatio-temporal patterns of $2\pi$-periodic solutions to second order reversibl... more The existence and spatio-temporal patterns of $2\pi$-periodic solutions to second order reversible equivariant autonomous systems with commensurate delays are studied using the Brouwer $O(2) \times \Gamma \times \mathbb Z_2$-equivariant degree theory. The solutions are supposed to take their values in a prescribed symmetric domain $D$, while $O(2)$ is related to the reversal symmetry combined with the autonomous form of the system. The group $\Gamma$ reflects symmetries of $D$ and/or possible coupling in the corresponding network of identical oscillators, and $\mathbb Z_2$ is related to the oddness of the right-hand side. Abstract results, based on the use of Gauss curvature of $\partial D$, Hartman-Nagumo type {\it a priori bounds} and Brouwer equivariant degree techniques, are supported by a concrete example with $\Gamma = D_8$ -- the dihedral group of order $16$.
First we characterize all the polynomial vector fields in $\R^4$ which have the Clifford torus as... more First we characterize all the polynomial vector fields in $\R^4$ which have the Clifford torus as an invariant surface. After we study the number of invariant meridians and parallels that such polynomial vector fields can have in function of the degree of these vector fields.
We analyze the dynamics of a 4-parameter family of planar ordinary differential equations, given ... more We analyze the dynamics of a 4-parameter family of planar ordinary differential equations, given by a polynomial of degree 5 that is equivariant under a symmetry of order 6. We obtain the number of limit cycles as a function of the parameters, and provide criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle surrounding either 1, 7 or 13 critical points, the origin being always one of these points. The method used is the reduction of the problem to an Abel equation.
Lotka-Volterra model is one of the most popular in biochemistry. It is used to analyze cooperativ... more Lotka-Volterra model is one of the most popular in biochemistry. It is used to analyze cooperativity, autocatalysis, synchronization at large scale and especially oscillatory behavior in biomolecular interactions. These phenomena are in close relationship with the existence of first integrals in this model. In this paper we determine the independent first integrals of a family of $n$--dimensional Lotka-Volterra systems. We prove that when $n=3$ and $n=4$ the system is completely integrable. When $n\geq6$ is even, there are three independent first integrals, while when $n\geq5$ is odd there exist only two independent first integrals. In each of these mentioned cases we identify in the parameter space the conditions for the existence of Darboux first integrals. We also provide the explicit expressions of these first integrals.
We analyze the dynamics of a class of Z2n-equivariant differential equations of the form ż = pzn−... more We analyze the dynamics of a class of Z2n-equivariant differential equations of the form ż = pzn−1z̄n−2 +sznz̄n−1− z̄2n−1, where z is complex, the time t is real, while p and s are complex parameters. This study is the generalisation to Z2n of previous works with Z4 and Z6 symmetry. We reduce the problem of finding limit cycles to an Abel equation, and provide criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle that surrounds either 1, 2n + 1 or 4n + 1 equilibria, the origin being always one of these points.
In this paper we analyze a generic dynamical system with D2 constructed via a Cayley graph. We st... more In this paper we analyze a generic dynamical system with D2 constructed via a Cayley graph. We study the Hopf bifurcation and find conditions for obtaining a unique branch of periodic solutions. Our main result comes from analyzing the system under weak coupling, where we identify the conditions for heteroclinic cycle between four equilibria in the two-dimensional fixed point subspace of some of the isotropy subgroups of D2 × S1. We also analyze the stability of the heteroclinic cycle.
Abstract. We analyze the dynamics of a class of Z6−equivariant sys-tems of the form z ̇ = pz2z ̄ ... more Abstract. We analyze the dynamics of a class of Z6−equivariant sys-tems of the form z ̇ = pz2z ̄ + sz3z̄2 − z̄5, where z is complex, the time t is real, while p and s are complex parameters. This study is the natural continuation of a previous work (M.J. Álvarez, A. Gasull, R. Prohens, Proc. Am. Math. Soc. 136, (2008), 1035–1043) on the normal form of Z4−equivariant systems. Our study uses the reduction of the equation to an Abel one, and provide criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle surrounding either 1, 7 or 13 critical points, the origin being always one of these points. Keywords: Planar autonomous ordinary differential equations, symmetric polinomial systems, limit cycles AMS Subject Classifications:
We analyze the dynamics of a class of Z2n-equivariant dif- ferential equations of the form u z = ... more We analyze the dynamics of a class of Z2n-equivariant dif- ferential equations of the form u z = pz n 1 ¯ z n 2 +sz nn 1 ¯ 2n 1 , where z is complex, the time t is real, while p and s are complex parameters. This study is the generalisation to Z2n of previous works with Z4 and Z6 symmetry. We reduce the problem of finding limit cycles to an Abel equation, and provide criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle that surrounds either 1, 2n+1 or 4n+1 equilibria, the origin being always one of these points.
We analyze the generating mechanisms for heteroclinic cycles in Z_2×Z_2×Z_2--equivariant ODEs, no... more We analyze the generating mechanisms for heteroclinic cycles in Z_2×Z_2×Z_2--equivariant ODEs, not involving Hopf bifurcations. Such cycles have been observed in particle physics systems with the mentioned symmetry, in absence of the Hopf bifurcation, see bury and Park, and as far as we know, there is no available theoretical data explaining these phenomena. We use singularity theory to study the equivalence in the group-symmetric context, as well as the recognition problem for the simplest bifurcation problems with this symmetry group. Singularity results highlight different mechanisms for the appearance of heteroclinic cycles, based on the transition between the bifurcating branches. On the other hand, we analyze the heteroclinic cycle of a generic dynamical system with the symmetry of the group Z_2×Z_2×Z_2 acting on a eight--dimensional torus T^8, constructed via a Cayley graph, under weak coupling. We identify the conditions for heteroclinic cycle between four equilibria in the ...
The existence and spatio-temporal patterns of $2\pi$-periodic solutions to second order reversibl... more The existence and spatio-temporal patterns of $2\pi$-periodic solutions to second order reversible equivariant autonomous systems with commensurate delays are studied using the Brouwer $O(2) \times \Gamma \times \mathbb Z_2$-equivariant degree theory. The solutions are supposed to take their values in a prescribed symmetric domain $D$, while $O(2)$ is related to the reversal symmetry combined with the autonomous form of the system. The group $\Gamma$ reflects symmetries of $D$ and/or possible coupling in the corresponding network of identical oscillators, and $\mathbb Z_2$ is related to the oddness of the right-hand side. Abstract results, based on the use of Gauss curvature of $\partial D$, Hartman-Nagumo type {\it a priori bounds} and Brouwer equivariant degree techniques, are supported by a concrete example with $\Gamma = D_8$ -- the dihedral group of order $16$.
First we characterize all the polynomial vector fields in $\R^4$ which have the Clifford torus as... more First we characterize all the polynomial vector fields in $\R^4$ which have the Clifford torus as an invariant surface. After we study the number of invariant meridians and parallels that such polynomial vector fields can have in function of the degree of these vector fields.
We analyze the dynamics of a 4-parameter family of planar ordinary differential equations, given ... more We analyze the dynamics of a 4-parameter family of planar ordinary differential equations, given by a polynomial of degree 5 that is equivariant under a symmetry of order 6. We obtain the number of limit cycles as a function of the parameters, and provide criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle surrounding either 1, 7 or 13 critical points, the origin being always one of these points. The method used is the reduction of the problem to an Abel equation.
Lotka-Volterra model is one of the most popular in biochemistry. It is used to analyze cooperativ... more Lotka-Volterra model is one of the most popular in biochemistry. It is used to analyze cooperativity, autocatalysis, synchronization at large scale and especially oscillatory behavior in biomolecular interactions. These phenomena are in close relationship with the existence of first integrals in this model. In this paper we determine the independent first integrals of a family of $n$--dimensional Lotka-Volterra systems. We prove that when $n=3$ and $n=4$ the system is completely integrable. When $n\geq6$ is even, there are three independent first integrals, while when $n\geq5$ is odd there exist only two independent first integrals. In each of these mentioned cases we identify in the parameter space the conditions for the existence of Darboux first integrals. We also provide the explicit expressions of these first integrals.
We analyze the dynamics of a class of Z2n-equivariant differential equations of the form ż = pzn−... more We analyze the dynamics of a class of Z2n-equivariant differential equations of the form ż = pzn−1z̄n−2 +sznz̄n−1− z̄2n−1, where z is complex, the time t is real, while p and s are complex parameters. This study is the generalisation to Z2n of previous works with Z4 and Z6 symmetry. We reduce the problem of finding limit cycles to an Abel equation, and provide criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle that surrounds either 1, 2n + 1 or 4n + 1 equilibria, the origin being always one of these points.
In this paper we analyze a generic dynamical system with D2 constructed via a Cayley graph. We st... more In this paper we analyze a generic dynamical system with D2 constructed via a Cayley graph. We study the Hopf bifurcation and find conditions for obtaining a unique branch of periodic solutions. Our main result comes from analyzing the system under weak coupling, where we identify the conditions for heteroclinic cycle between four equilibria in the two-dimensional fixed point subspace of some of the isotropy subgroups of D2 × S1. We also analyze the stability of the heteroclinic cycle.
Uploads
Papers by Adrian Calin Murza