Homogeneous Dynamics in the Plane

This chapter considers autonomous dynamic systems in the normal (explicit) form: x ̇ = F ( x , y ) , y ̇ = G ( x , y ) , ( x , y ) ∈ R 2 $$ \dot {\mathbf {x}}\mathbf {= F(x,y),\,} \dot {\mathbf {y}}\mathbf {= G(x,y),}\, (x,y)\in {\mathbf {R}}^2$$ , F ( 0 , 0 ) = 0 , G ( 0 , 0 ) = 0 $$F(0,0)=0, G(0,0)=0$$ , where the C 1 $${C^1}$$ -class functions, F $$\mathbf {F}$$ and G $$\mathbf {G}$$ , are homogeneous of the same degree, m ∈ R $$m\in \mathbf {R}$$ . The C 1 $$C^1$$ -class assumption about the governing functions, F $$\mathbf {F}$$ , G $$\mathbf {G}$$ , ensures everywhere the existence and uniqueness of the coordinate solutions: [ x , y ] = [ φ 1 ( t ) , φ 2 ( t ) ] = φ ( t ) $$[\,x,y\,]\,=\,[\,\boldsymbol {\varphi }_{\mathbf {1}}(\mathbf {t}),\boldsymbol {\varphi }_{\mathbf {2}}(\mathbf {t})\,]\,=\,\boldsymbol {\varphi }(\mathbf {t})$$ . By homogeneity property and by using the ratio variable, r = y ∕ x , x ≠ 0 $$\mathbf {r}=y/x,\, x\neq 0$$ , we rewrite the dynamic system as: x ̇ = x | x | m −1 f ( r ) , y ̇ = x | x | m −1 g ( r ) , x ≠ 0 $$\dot {\mathbf {x}}=x| x|{ }^{m-1}\mathbf {f(r)},\, \dot {\mathbf {y}}=x| x|{ }^{m-1}\mathbf {g(r)},\, x\neq 0$$ . The ratio of the individual coordinate solutions, r = ρ ( t ) = φ 2 ( t ) ∕ φ 1 ( t ) , $$\mathbf {r}=\rho (\boldsymbol {t})=\varphi _{2}(t)/\varphi _{1}(t),$$ has the time derivative: r ̇ = dr ∕ dt = | x | m −1 h ( r ) $$\dot {\mathbf {r}}=dr/dt=|x|{ }^{m-1}\mathbf {h(r)}$$ , where, h ( r ) = g ( r ) − rf ( r ) , $$\mathbf {h(r)}=g(r)-rf(r),$$ is called a director function; its roots, h ( α ) = 0 $$\mathbf {h}\,(\alpha ) =\mathbf{0}$$ , are called director roots; rays, y = α x $$\mathbf {y}=\alpha \,\mathbf{x}$$ , are called directrices; and f ( α ) $$f(\alpha )$$ is called a directrix value. A directrix reflects a constant ratio solution, i.e., ρ ( t ) = α , φ 2 ( t ) = α φ 1 ( t ) $$\rho \,(\mathbf {t}) =\alpha ,\, \varphi _{2}(t)=\alpha \,\varphi _{1}(t)$$ . Introducing the primitive H ( r ) $$H(r)$$ , defined as H ( r ) = ∫ f ( r ) h ( r ) dr $$\mathbf {H(r)}=\int \frac {f(r)}{h(r)}\,dr$$ , where the intervals for, H ( r ) $$H(r)$$ , correspond to intervals given by the director roots, ( α ) $$(\alpha )$$ . By r ̇ $$\dot {\mathbf {r}}$$ and x ̇ $$\dot {\mathbf {x}}$$ it is shown that the nonstationary ratio solution, ρ ( t ) $$\rho (\mathbf {t})$$ , must solve the fundamental differential equation: r ̇ = Q ( r ) = | k 0 exp { H ( r ) } | m −1 h ( r ) $$\dot {\mathbf {r}}=\mathbf {Q(r)}=|k_{0}\exp \{H(r)\}|^{m-1}h(r)$$ . Thus dynamics of any autonomous system of two interdependent homogeneous differential equations—having same degree of homogeneity—is essentially given succinctly through a single autonomous differential equation. Having obtained, ρ ( t ) $$\rho \,(\mathbf {t})$$ , corresponding coordinate solutions, φ 1 ( t ) $$\varphi _{1}(t)$$ , φ 2 ( t ) $$\varphi _{2}(t)$$ , are then given as: φ 1 ( t ) = k 0 exp { H [ ρ ( t ) ] } , φ 2 ( t ) = ρ ( t ) φ 1 ( t ) $$ \boldsymbol {\varphi }_{\mathbf {1}}(\mathbf {t})=k_{0}\,\exp \{H[\,\rho (t)\,]\},\, \boldsymbol {\varphi }_{\mathbf {2}}(\mathbf {t})=\boldsymbol {\rho }(\mathbf {t})\,\boldsymbol {\varphi }_{\mathbf {1}}(\mathbf {t})$$ , and the trajectories as loci are given by the equation: x − k 0 exp { H ( y ∕ x ) } = 0 $$\mathbf {x}-{\mathbf {k}}_{\mathbf {0}}\exp \{H(\mathbf {y}/\mathbf {x})\} = 0$$ . Thus, we have obtained the complete set of solutions to the homogeneous dynamic system. Usually, the nonstationary ratio and coordinate solutions cannot be expressed in closed form (finite combination of elementary functions), but the fact that these solutions are obtained in terms of well-defined primitives [ integrands being determined by the governing functions: f $$\mathbf {f}$$ , g $$\mathbf {g}$$ ] will allow us to deduce the qualitative behavior of the family of solutions to for any specification of the basic governing functions: F $$\mathbf {F}$$ , G $$\mathbf {G}$$ . The global asymptotic ratio stability conditions follow from, h ( r ) $$\mathbf {h(r)}$$ , and are stated in Theorem 28.4, weaker ratio stabilities in Theorem 28.5. The trajectory geometry is given in Theorem 28.8. The stability properties of the coordinate solutions are summarized in Theorems 28.9–28.12.