11 problems

Let $\mathcal{A}$ denote the attractor describing generic behaviour near the initial singularity, and let $\mathrm{K}^{\circ}$, $\mathcal{T}_{R_1}$, $\mathcal{T}_{R_3}$, and…

The dynamical system has plane-wave fixed points $\mathrm{PW}^{\pm}$, a fixed point source $\mathrm{RT}$, a fixed point $\mathrm{PW}^{0}$, a fixed point $\mathrm{T}_3$, and variabl…

Let $M_0^{\lambda,m,n}$ and $M_1^{\lambda,m,n}$ be the equilibria of the dynamical system, with $W^u(M_0)$ denoting the unstable manifold of $M_0$ and $W^s(M_1)$ denoting the stabl…

Let $(u_{C_1},w_{C_1})$, $(u_{C_2},w_{C_2})$, and $(u_W,w_W)$ be the equilibria of system … possesses heteroclinic orbits connecting $(u_{C_2},w_{C_2})$ to $(u_{C_1},w_{C_1})$ and…

Let $(\bar{p},s)$ be parameters for the fast subsystem, with three hyperbolic equilibria $x_l$, $x_m$, and $x_r$, and let heteroclinic connections be detected by intersections of t…

Generic simplicity conjecture. There is a generic subset (a Baire subset) of choices for $\phi$ in $$ such that, if $u$ is a heteroclinic orbit, all of the eigenvalues of $H(t)$ ar…

Let $E$ be the space of heteroclinic orbits of … , … Thus the Floer complex should compute the homology of the space of heteroclinic orbits. The source presents this as a reasonabl…

Let $C_k(R)$ be the free $R$-module generated by the $k$-dimensional connecting manifolds of … , there are maps … such that each $\partial_k$ is an $R$-module homomorphism,…

Let $H(t)$ denote the operator whose eigenvalues are considered along a heteroclinic orbit $u$ of … such that, for every heteroclinic orbit $u$, all eigenvalues of $H(t)$ are simpl…

Let $\mathcal{H}$ be the space of heteroclinic orbits of the flow associated with … is restricted to $\mathcal{H}$, all equilibria become nondegenerate critical points in the sense…

Let $u$ be an eternal solution of the semilinear parabolic equation \, with coefficients $a_i$ that decay sufficiently fast as $|x|\to\infty$. Heteroclinic-orbit conjecture. All…