J0 = 
{{0,1},
{1,μ}};

esys = Eigensystem[J0];

unstablev = esys[[2,2]];
stablev = esys[[2,1]];

μsub = {μ->-0.9};  (* critical is -0.8645 *)
ϵ = 0.001;
tmax = 
Which[
(μ/.μsub)<-0.8645,100,
True,16
];
sol =
NDSolve[
{x'[t] == y[t],
y'[t] == μ*y[t] + x[t] - x[t]^2 + x[t]*y[t],
x[0] == ϵ*unstablev[[1]],
y[0] == ϵ*unstablev[[2]]
}/.μsub,
{x,y},
{t,tmax}
];
NEplot =
ParametricPlot[
Evaluate[{x[t],y[t]}/.sol],{t,0,tmax},
PlotStyle->Red
];


tmax = 
Which[
(μ/.μsub)<-0.8645,10,
True,10
];
sol =
NDSolve[
{x'[t] == y[t],
y'[t] == μ*y[t] + x[t] - x[t]^2 + x[t]*y[t],
x[0] == -ϵ*unstablev[[1]],
y[0] == -ϵ*unstablev[[2]]
}/.μsub,
{x,y},
{t,tmax}
];
SWplot =
ParametricPlot[
Evaluate[{x[t],y[t]}/.sol],{t,0,tmax},
PlotStyle->Red
];

tmax = 4.2;
sol =
NDSolve[
{-x'[t] == y[t],
-y'[t] == μ*y[t] + x[t] - x[t]^2 + x[t]*y[t],
x[0] == ϵ*stablev[[1]],
y[0] == ϵ*stablev[[2]]
}/.μsub,
{x,y},
{t,tmax}
];
NWplot =
ParametricPlot[
Evaluate[{x[t],y[t]}/.sol],{t,0,tmax},
PlotStyle->Blue
];

tmax = 
Which[
(μ/.μsub)<-0.8645,10,
True,20
];
sol =
NDSolve[
{-x'[t] == y[t],
-y'[t] == μ*y[t] + x[t] - x[t]^2 + x[t]*y[t],
x[0] == -ϵ*stablev[[1]],
y[0] == -ϵ*stablev[[2]]
}/.μsub,
{x,y},
{t,tmax}
];
SEplot =
ParametricPlot[
Evaluate[{x[t],y[t]}/.sol],{t,0,tmax},
PlotStyle->Blue
];

tmax = 
Which[
(μ/.μsub)<-0.8645,50,
True,0.001
];
sol =
NDSolve[
{x'[t] == y[t],
y'[t] == μ*y[t] + x[t] - x[t]^2 + x[t]*y[t],
x[0] == 1.1,
y[0] == 0
}/.μsub,
{x,y},
{t,tmax}
];
spiralplot =
ParametricPlot[
Evaluate[{x[t],y[t]}/.sol],{t,0,tmax},
PlotStyle->Purple
];

Show[
NEplot,SWplot,NWplot,SEplot,spiralplot,
PlotRange->All
]