# Glossary of Dynamical Systems Terms

• Asymptotic stability A fixed point is asymptotically stable if it is stable and nearby initial conditions tend to the fixed point in positive time. For limit cycles , it is called orbital asymptotic stability and then there is an associated phase shift. A fixed point is locally stable if the eigenvalues of the linearized system have negative real parts. A limit cycle is orbitally asymptotically stable if the Floquet multipliers of the linearized system lie inside the unit circle with the exception of a multiplier with value 1.
• Attractor An attractor is a trajectory of a dynamical system such that initial conditions nearby it will tend toward it in forward time. Often called a stable attractor but this is redundant.
• Averaging A method in which one can average over the period of some system when one of the variables evolve slowly compared to length of the period.
• Bifurcation point This is a point in parameter space where we can expect to see a change in the qualitative behavior of the system, such as a loss of stability of a solution or the emergence of a new solution with different properties.
• Bifurcation diagram This is a depiction of the solution to a dynamical system as one or more parameters vary. Typically, the horizontal axis has the parameter and the vertical axis has some aspect of the solution, such as, the norm of the solution, the maximum and/or minimum values of one of the state variables, the frequency of a solution, or the average of one of the state variables.
• Bistability The presence of two coexistent attractors in a dynamical system. For example, two stable fixed points or a stable fixed point and a stable limit cycle. Birhymicity has been used to define a system with two stable limit cycles.
• Chaos A long time behavior of a dynical system characterized by a great deal of irregularity. Typically, two nearby trajectories will diverge exponentially.
• Continuation branch A curve of fixed points, limit cycles, etc as a function of some parameter.
• Domain of attraction Also called basin of attraction is the region in state space of all initial conditions that tend to a particular solution such as a limit cycle, fixed point, or other attractor.
• Eigenvalues Complex numbers, that satisfy where A is an matrix and x is some vector. In this case, x is called an eigenvector.
• Fixed Point A special trajectory of the dynamical system which does not change in time. It is also called an equilibrium, steady-state, or singular point of the system.
• Floquet Multipliers Complex numbers, that satisfy where is an periodic matrix and is a periodic vector.
• Global stability When referring to an attractor, this means that the domain of attraction is the entire phase space. When referring to a dynamical system, it means that all initial conditions tend to one of the attractors.
• Hard loss of stability This term is often used when there is a subcritical bifurcation which turns around and stabilizes. Most typically, it is used in reference to a Hopf bifurcation. As some parameter is changed, the switch in behavior is sudden rather than gradual. See also Hysteresis.
• Heteroclinic orbit A trajectory which tends to fixed points in both positive and negative time. The fixed points do not have to be the same.
• Hopf bifurcation This is the appearance of a new branch of periodic solutions from a branch of fixed points. The classic earmark of this is that a pair of complex conjugate eigenvalues of the linearized system change from negative to positive real parts.
• Homoclinic A trajectory that tends to the same fixed point in positive and negative time.
• Hysteresis As a parameter is increased, the behavior makes a sudden jump at a particular value of the parameter. But as the parameter is then decreased, the jump back to the original behavior does not occur until a much lower value. In the region between the two jumps, the system, is bistable.
• Invariant set Any orbit or region in phase space which does not change under the differential equations. For example, a fixed point or a periodic orbit are both invariant sets.
• Limit cycle An isolated periodic orbit (as compared to the solutions to for which there are infinately many periodic solutions passing through any region in phase space.)
• Limit point A saddle-node bifurcation for fixed points. Two fixed points coalesce and disappear as a parameter varies. Also called a turning point.
• Linearization If the dynamical system is then the linearization is the dynamical system, where A is the derivative of F evaluated at a solution.
• Local stability A fixed point is locally stable or stable if nearby initial conditions remain nearby in positive time.
• Liapunov function Any function, V(X) of n variables which vanishes at a fixed point, say, X=0 , is positive in some neighborhood of the fixed point, and satisfies dV/dt < 0 along trajectories of the dynamical system.
• Nullclines Curves drawn in the phase portrait along which one of the state variables does not change in time. Used frequently in 2-dimensional systems.
• Period doubling bifurcation A bifurcation in which the system switches to a new behavior with twice the period of the original system. The hallmark of this is a Floquet multiplier of -1.
• Period doubling cascade A sequence of period doubling bifurcations ultimately ending in chaotic behavior.
• Periodic orbit A trajectory which after time , the period, comes back to its initial point.
• Phase plane A two-dimensional phase portrait of a two-dimensional dynical system.
• Phase portrait A plot of two or more dynamical variables against each other.
• Phase space The set of all possible initial conditions for a dynamical system. The dimension of the phase space is the number of intial conditions required to uniquely specify a trajectory; it is the number of variables in the dynamical system.
• Orbit Trajectory of differential equation.
• Saddle-loop A bifurcation in which a limit cycle collides with a saddle-point at a homoclinic orbit. As the bifurcation is approached, the period of the limit cycle tends to infinity as .
• Saddle-node loop A bifurcation in which a limit cycle collides with a saddle-node point. As the bifurcation is approached, the period of the limit cycle tends to infinity as the .
• Saddle-node point A limit point.
• Saddle point A fixed point that has at least one positive eigenvalue and one negative eigenvalue in its linearization. More generally, a fixed point for which there are trajectories that tend to the fixed point in both positive and negative time.
• Sink A locally asymptotically stable fixed point.
• Source A fixed point that is unstable and not a saddle-point. All trajectories move away in positive time.
• Stable manifold The set of all points in phase space which are attracted to a fixed point or other invariant orbit in positive time .
• Subcritical bifurcation The branch of solutions occurs for parameter values on the opposite side of the loss of stability. Often indicates that the new branch is unstable.
• Supercritical bifurcation The branch of solutions occurs for parameter values on the same side as the loss of stability. Often indicates that the new branch is stable.
• Trajectory The solution to a dynamical system in forward and backward time passing through a specified initial condition.
• Unstable manifold The set of all points in phase space which are attracted to a fixed point or other invariant orbit in negative time .