Good question! This is really a question of definitions. As you observe, marginally stable systems can produce sustained oscillations! However, the amplitude of these oscillations depends on the initial condition, meaning that small perturbations to the initial condition can lead to a 'different' oscillation. It is this small difference that means that we don't call this a limit cycle. A limit cycle is an oscillation of fixed shape, which nearby trajectories will tend towards (or away from if the limit cycle is unstable). The idea is to try to extend the idea of a stable point, to a stable periodic behaviour. For example, think of your heartbeat. This is a periodic behaviour which in steady state has a fixed amplitude and frequency. In order to maintain this behaviour, your body must be able to make small adjustments in response to perturbations to keep the amplitude a frequency constant. We then introduce the concept of a limit cycle to describe this behaviour mathematically, and then generalise the ideas of stability or instability of a point to apply in this setting as well!
@@tech_science_tutos4155 a Limit cycle is a phenomenal that only occurs in non linear systems! a marginal stable oscillation is not a limit cycle! these are 2 different things! a limit cycle is of fixed amplitude and its amplitude is independent of initial values (where you start the sytem), therfore leading to always a single tragectory....anywhere nearby you start, you are always connected back to this tragectory! a marginaly stable or unstable ossilation is depended of varying aplituded as its amplitude is depends on initial value...therfore leading to many tragectory nearby! in addition to limit cycle or limit osilation, the following behaviours are also related to only non linear system: 1. Finite escape time: the state of a linear system can only go to infinity as the time aproaches infinite...however in nonlinear systems the state can go to infinity in a limited time (i.e. y=1/(t-2)). here y can go to infinite in a limited time (t=2) 2. multiple isolated equilibria: a linear system can have only one isolated equlibrium point, a nonlinear system however can have One, multipe or infinitly many isolated equilibirium points...equilibirum point is a point in which sytem rests (system drivate is 0)!!! it losses all its energy.. i.e y_drivative=t^2...here t=0 is an equilibrium point. 3. input depending stability: the stablility of a equilibrium point is not always a system property (as it is the case for LTI linear time iindependent systems), but can be input-dependent. so a certain limited input value may cause instablilty...i.e y=1/(t-2), here t=2 will cause y=infinite which is unstalbe. 4. irregular oscillations: a linear system responds to a sinusoidal exicattion with same frequency...but for nonlinear system ossilation of different frequencies can occur! 5. limit cycle: marginal stable oscillations can exist in linear systems only theoretically!!!! because they are extremely parameter-sensitive and their amplitude depends on initial state values or where the system starts from (0,0 or 1,2).... a stalbe limit cycle in nonlinear systems is of fixed amplitude and fixed frequency, independent of the initial state and quite robuts against parameter variatiosn or disturbances! 6. Chaos: A nonlinear system can have a more complicated steady-state behavior that is neither equilibrium, nor periodic or almost-periodic oscillation. It is irregular and exhibits some “stochastic” characters, despite the deterministic nature of the system, i.e., the trajectories never settle down to fixed points or to period orbits. Such behavior is usually referred to as chaos 7. Multiple modes of behavior: While a linear system can only show some fixed characteristic mode (periodic oscillation, exponential decay etc.) at once, it is not unusual for two or more modes of behavior to be exhibited by the same nonlinear system with continuous or even discontinuous change-overs between the modes try out phase Portrait in matlab for van der pol equation...there you can easily visualized both limit cycle (limit ossilation) and periodic ossilations (marginal ossialtions) thank you
Thank u))
can a linear system produce a limte cycle(marginal stability case)?
Good question! This is really a question of definitions. As you observe, marginally stable systems can produce sustained oscillations! However, the amplitude of these oscillations depends on the initial condition, meaning that small perturbations to the initial condition can lead to a 'different' oscillation. It is this small difference that means that we don't call this a limit cycle. A limit cycle is an oscillation of fixed shape, which nearby trajectories will tend towards (or away from if the limit cycle is unstable). The idea is to try to extend the idea of a stable point, to a stable periodic behaviour. For example, think of your heartbeat. This is a periodic behaviour which in steady state has a fixed amplitude and frequency. In order to maintain this behaviour, your body must be able to make small adjustments in response to perturbations to keep the amplitude a frequency constant. We then introduce the concept of a limit cycle to describe this behaviour mathematically, and then generalise the ideas of stability or instability of a point to apply in this setting as well!
@@richard_pates got it. thanks a lot for ur detailed response.
@@tech_science_tutos4155
a Limit cycle is a phenomenal that only occurs in non linear systems!
a marginal stable oscillation is not a limit cycle! these are 2 different things!
a limit cycle is of fixed amplitude and its amplitude is independent of initial values (where you start the sytem), therfore leading to always a single tragectory....anywhere nearby you start, you are always connected back to this tragectory!
a marginaly stable or unstable ossilation is depended of varying aplituded as its amplitude is depends on initial value...therfore leading to many tragectory nearby!
in addition to limit cycle or limit osilation, the following behaviours are also related to only non linear system:
1. Finite escape time: the state of a linear system can only go to infinity as the time aproaches infinite...however in nonlinear systems the state can go to infinity in a limited time (i.e. y=1/(t-2)). here y can go to infinite in a limited time (t=2)
2. multiple isolated equilibria: a linear system can have only one isolated equlibrium point, a nonlinear system however can have One, multipe or infinitly many isolated equilibirium points...equilibirum point is a point in which sytem rests (system drivate is 0)!!! it losses all its energy.. i.e y_drivative=t^2...here t=0 is an equilibrium point.
3. input depending stability: the stablility of a equilibrium point is not always a system property (as it is the case for LTI linear time iindependent systems), but can be input-dependent.
so a certain limited input value may cause instablilty...i.e y=1/(t-2), here t=2 will cause y=infinite which is unstalbe.
4. irregular oscillations: a linear system responds to a sinusoidal exicattion with same frequency...but for nonlinear system ossilation of different frequencies can occur!
5. limit cycle: marginal stable oscillations can exist in linear systems only theoretically!!!!
because they are extremely parameter-sensitive and their amplitude depends on initial state values or where the system starts from (0,0 or 1,2)....
a stalbe limit cycle in nonlinear systems is of fixed amplitude and fixed frequency, independent of the initial state and quite robuts against parameter variatiosn or disturbances!
6. Chaos: A nonlinear system can have a more complicated steady-state behavior that is neither
equilibrium, nor periodic or almost-periodic oscillation. It is irregular and exhibits some “stochastic”
characters, despite the deterministic nature of the system, i.e., the trajectories never settle down
to fixed points or to period orbits. Such behavior is usually referred to as chaos
7. Multiple modes of behavior: While a linear system can only show some fixed characteristic mode
(periodic oscillation, exponential decay etc.) at once, it is not unusual for two or more modes of
behavior to be exhibited by the same nonlinear system with continuous or even discontinuous
change-overs between the modes
try out phase Portrait in matlab for van der pol equation...there you can easily visualized both limit cycle (limit ossilation) and periodic ossilations (marginal ossialtions)
thank you